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Some further discussion on $K_1$ of elliptic curves over number fields can be found in the answers to this MO-question. As noted there, there is a recent paper of Kondo and Yasuda, in which they provide some computations of $K_1$ of elliptic curves over function fields. I think their computations would imply a complete calculation of $V(E)$ of an elliptic curve $E$ provided the Parshin conjecture is true.

Some further discussion on $K_1$ of elliptic curves over number fields can be found in the answers to this MO-question. As noted there, there is a recent paper of Kondo and Yasuda, in which they provide some computations of $K_1$ of elliptic curves over function fields. I think their computations would imply a complete calculation of $V(E)$ of an elliptic curve $E$ provided the Parshin conjecture is true.

Finite fields: For smooth projective curves over finite fields (resp. their algebraic closures), there are some K-theory computations in this paper of Coombes:

• K. Coombes. On the K-theory of curves over finite fields. JPAA 51 (1988) 79-87.

From Theorem 1 in that paper, the Jacobian part is trivial. There are also two conjectures in the paper which would imply that the same formula holds over finite fields, and not just their algebraic closures. I am not sure about the status of these conjectures, maybe they follow from the Quillen-Lichtenbaum conjecture and thus ultimately from Voevodsky's work. For smooth affine curves over finite fields, the work of Harder (G. Harder. Die Kohomologie $S$-arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 42 (1977), 135--175) shows that $SK_1(k[C])$ is torsion. In the situation at hand, this probably also follows using the localization sequence from Coombes result.

Finite fields: Everything is known in the case of smooth curves over finite fields, cf. Theorems VI.6.4 and VI.6.7 of Weibel's $K$-book. The "Jacobian part" is trivial, so that $K_1(C)\cong\mathbb{F}_q^\times\oplus\mathbb{F}_q^\times$ for $C$ a smooth projective curve over $\mathbb{F}_q$. The localization sequence allows to deal with arbitrary smooth curves over $\mathbb{F}_q$.

As is already apparent from Steven Landsburg's answer, the question is not as elementary as one might think at first glance. Already for $K_1$ of curves, there are a couple of as yet unresolved problems. This is mainly due to the fact that the localization sequence relates the K-theory of curves over function fields to the K-theory of their (higher-dimensional) models. So let me outline some known results and open problems concerning $K_1$ of curves, the case distinction being in terms of the base field.

From Theorem 1 in that paper, you find $K_1(C)\cong K_1(k)\oplus K_1(k)$ where $C$ is a smooth projective curve over $k=\overline{\mathbb{F}_p}$. There are also two conjectures in the paper which would imply that the same formula holds over finite fields, and not just their algebraic closures. I do not know if these conjectures have been resolved by now.

For smooth affine curves over finite fields, the following paper of Harder is relevant:

• G. Harder. Die Kohomologie $S$-arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 42 (1977), 135--175.

The results proved in that paper imply that $SK_1(k[C])=\ker\left(\det:K_1(k[C])\to k^\times\right)$ is torsion. However, I do not know of further concrete calculations.

Global fields: For curves over number fields, there are conjectures of Vaserstein and Bloch. Vaserstein's conjecture asks if $SK_1(C)$ is torsion if $C$ is a smooth affine curve over a number field. Bloch's conjecture asks if $V(C)=\ker\left(CH^2(C,1)\to K_1(k)\right)$ is torsion for $C$ a smooth projective curve over a number field $k$. Both conjectures are equivalent by a result of Raskind:

You can find statements of the conjectures, references to the relevant papers and the proof of the equivalence in that paper. The function field situation is not discussed so explicitly in that paper, but maybe something similar is true there... Note that Bloch's conjecture equivalently can be stated as $K_1(C)\cong K_1(k)\oplus K_1(k)$ where $C$ is a smooth projective curve over $k=\overline{\mathbb{Q}}$.

Some further discussion on $K_1$ of elliptic curves over number fields can be found in the answers to this MO-question. As noted there, there is a recent paper of Kondo and Yasuda, in which they provide some computations of $K_1$ of elliptic curves over function fields. I think their computations would imply that $V(E)$ of an elliptic curve $E$ over a function field is torsion, of explicitly computed order, provided the Parshin conjecture is true (which would imply finite-generation and thus vanishing of the uniquely divisible part).

Generic fields: Finally, the statements outlined above say that $K_1$ of curves over "small fields" can be understood and is not too big (although this depends on serious conjectures). In general, this would appear to fail rather badly. There is a preprint of Rosenschon and Srinivas in which they show that for $k\subsetneq K$ an extension of algebraically closed fields of characteristic $0$ and $C$ a sufficiently generic smooth projective curve over $K$, the product map from K-theory $k^\times\otimes\operatorname{Pic}(C)\to CH^2(C,1)$ is injective, so that $K_1(C)$ is quite a lot bigger than just $K^\times \oplus K^\times$.

ps: To also say something about your question (2), the tame symbol appears as a map in a colimit of localization sequences for K-theory. Take the curve $C$, finitely many points as closed subscheme $Z$ and the open complement $U$ - this yields a localization sequence. Taking the limit over larger and larger closed subschemes $Z$ gives a localization sequence connecting the K-theory of the curve $C$, the K-theory of its function field $k(C)$, and the direct sum of K-theories of the residue fields of the closed points of $C$. The tame symbol is then the boundary map $K_2(k(C))\to\bigoplus_{\kappa\in C^{(1)}} \kappa(x)^\times$.

As is already apparent from Steven Landsburg's answer, the question is not as elementary as one might think at first glance. Already for $K_1$ of curves, there are a couple of as yet unresolved problems. This is mainly due to the fact that the localization sequence relates the K-theory of curves over function fields to the K-theory of their (higher-dimensional) models.

Concerning (1): Let me discuss a bit what is known about the structure of $K_1$ of curves.

Generalities: Some of the structure of $K_1$ becomes clearer when looking at motives or motivic cohomology. If $C$ is a smooth projective curve over a field $k$, then there is a splitting of the motive (Chow or Voevodsky, does not matter): $$ M_k(C)\cong \mathbb{Z}_k\oplus \operatorname{Jac}(C/k)\oplus\mathbb{Z}_k(1)[2]. $$ Using Bott periodicity of algebraic K-theory, the two summands $\mathbb{Z}$ and $\mathbb{Z}(1)[2]$ each contribute one copy of $K_1(k)$ to $K_1(C)$. The remaining part of $K_1(C)$ comes from the motive $\operatorname{Jac}(C/k)$, which as the notation suggests comes from the Jacobian.

There are several ways to define groups related to this Jacobian part of $K_1(C)$. Rationally, we can use the Adams eigenspace decomposition $$ K_1(C)_{\mathbb{Q}}\cong\bigoplus_{q\geq 0} H^{2q-1,q}(C,\mathbb{Q}). $$ Known vanishing results for motivic cohomology, imply that the summands are trivial unless $q=1,2$. One copy of $K_1(k)$ is the weight one part ($q=1$), and the weight two part ($q=2$) is the combination of the other $K_1(k)$ and the stuff coming from the Jacobian motive. There are integral ways of defining the weight two part: we have $H^{3,2}(C,\mathbb{Z})=CH^2(C,1)=\ker(K_1(C)\to K_1(k(C))$, and Bloch defined the group $V(C)=\ker(CH^2(C,1)\to K_1(k))$. This is therefore a group which encodes integrally the stuff in $K_1(C)$ coming from the Jacobian motive. This also is the part of $K_1(C)$ that is not yet understood, and subject to several conjectures.

[The situation of smooth affine curves can be dealt with using the localization sequence, reducing to the projective curves. The analoguous thing to $V(C)$ in the case of smooth affine curves is $SK_1(k[C])=\ker(\det:K_1(k[C])\to K_1(k))$, which alternatively is the abelianization of $SL_\infty(k[C])$. That this group can be rather non-trivial is in the answer of Steven Landsburg.]

Now I will outline some known results and open problems concerning the "Jacobian part" of $K_1$ of curves, the case distinction being in terms of the base field.

From Theorem 1 in that paper, the Jacobian part is trivial. There are also two conjectures in the paper which would imply that the same formula holds over finite fields, and not just their algebraic closures. I am not sure about the status of these conjectures, maybe they follow from the Quillen-Lichtenbaum conjecture and thus ultimately from Voevodsky's work. For smooth affine curves over finite fields, the work of Harder (G. Harder. Die Kohomologie $S$-arithmetischer Gruppen über Funktionenkörpern. Invent. Math. 42 (1977), 135--175) shows that $SK_1(k[C])$ is torsion. In the situation at hand, this probably also follows using the localization sequence from Coombes result.

Global fields: For curves over number fields, there are conjectures of Vaserstein and Bloch for curves over number fields. Vaserstein's conjecture asks if $SK_1(C)$ is torsion if $C$ is a smooth affine curve, Bloch's conjecture asks if $V(C)$ is torsion for $C$ smooth projective. Both conjectures are equivalent by a result of Raskind:

You can find statements of the conjectures, references to the relevant papers and the proof of the equivalence in that paper. These conjectures state that the part of $K_1(C)$ coming from the Jacobian is torsion (and hence trivial over the algebraic closure of a global fields). Though the function field case is not discussed in this paper, probably similar statements would be consequences of Parshin's conjecture.

Some further discussion on $K_1$ of elliptic curves over number fields can be found in the answers to this MO-question. As noted there, there is a recent paper of Kondo and Yasuda, in which they provide some computations of $K_1$ of elliptic curves over function fields. I think their computations would imply a complete calculation of $V(E)$ of an elliptic curve $E$ provided the Parshin conjecture is true.

Generic fields: Finally, the statements outlined above say that $K_1$ of curves over "small fields" can be understood and is not too big (although this depends on serious conjectures). In general, this would appear to fail rather badly. There is a preprint of Rosenschon and Srinivas in which they show that for $k\subsetneq K$ an extension of algebraically closed fields of characteristic $0$ and $C$ a sufficiently generic smooth projective curve over $K$, the product map from K-theory $k^\times\otimes\operatorname{Pic}(C)\to CH^2(C,1)$ is injective, so that the Jacobian part of $K_1(C)$ is quite huge.

Concerning (2): To also say something about your question (2), the tame symbol appears as a map in a colimit of localization sequences for K-theory. Take the curve $C$, finitely many points as closed subscheme $Z$ and the open complement $U$ - this yields a localization sequence. Taking the limit over larger and larger closed subschemes $Z$ gives a localization sequence connecting the K-theory of the curve $C$, the K-theory of its function field $k(C)$, and the direct sum of K-theories of the residue fields of the closed points of $C$. The tame symbol is then the boundary map $K_2(k(C))\to\bigoplus_{\kappa\in C^{(1)}} \kappa(x)^\times$.