First, I am not sure how appropriate it is to answer my own question. If this is against the protocol here, let me know, and I'll promptly delete this answer and copy it as an addendum instead.
Here is an affirmative solution to the first question under the special assumption that, at some place $v_0$ of $K$, the rational function $g \in K(t) \cap K[[t]]$ has a unique (simple) pole of strictly maximum absolute value. This generalizes a result on power sums by D. Cantor (Canad. J. Math., 1969). The analytic argument involved is based on the Poincare-Lelong equation and is, in miniature, a technique from J.-B. Bost's proof of an important special case of the non-linear generalization by Ekedahl, Shepherd-Barron, and Taylor of the "$p$-curvature conjecture" of Grothendieck and Katz. (Algebraic leaves of algebraic foliations over number fields, IHES, 2001). In Bost's paper, the technique is applied under the so called Liouville property at the place $v_0$, roughly corresponding to the dominant pole assumption in the mini-problem which I consider here. (One might say that the argument here is just a straightforward application of the maximum principle.)
It seems that the remaining questions may be settled similarly under a dominant pole assumption. I have no idea for the general case, however; to me, the question is open even in the case $K = \mathbb{F}_q[x]$ and $f,g \in K(t)$.
Assumptions. For concreteness, I will assume that $K$ is a number field and the place $v_0$ is Archimedean. We have $f = \sum_{n \geq 0} a_nt^n \in K[[t]]$ algebraic, $g = \sum_{n \geq 0} b_nt^n \in K(t) \cap K[[t]]$ rational with all its poles in $K$, $g$ has a unique pole $\beta$ with $|\beta|_{v_0}$ maximal, and this pole is simple. We may thus write
$$
b_n = \beta^n \big( 1 + \sum_{i=1}^k q_i(n)\gamma_i^n \big), \quad q_i \in K[x], \, \textrm{ all } |\gamma_i|_{v_0} < 1.
$$
Notation. I write $\big( \sum \alpha_nt^n \big) * \big( \sum_n \beta_nt^n \big) := \sum_n \alpha_n\beta_nt^n$ for the Hadamard product. For $v$ a finite place of $K$, $|\cdot|_v$ will denote the absolute value in the class of $v$ with the normalization
$$
\log{|\cdot|_v} := \frac{\mathrm{ord}_v(\cdot)}{[K:\mathbb{Q}]}\log{|k(v)|}.
$$
For $v$ an infinite place corresponding to a complex embedding $\sigma : K \hookrightarrow \mathbb{C}$, let $|\cdot|_v := |\sigma(\cdot)|^{[K_v:\mathbb{R}]/[K:\mathbb{Q}]}$. The absolute logarithmic Weil height of a point $\mathbf{x} = (x_i)_i$ of a projective space is then $h(\mathbb{x}) := \sum_v \max_j \log{|x_j|_v}$. For $x \in \overline{\mathbb{Q}}$, also write $h(x) := h(1:x)$, viewing $x$ as the point $(1:x)$ on $\mathbb{P}^1$. For $F = \sum_{n \geq 0} a_nt^n \in K[[t]]$, denote $F_{/N} = \sum_{n =0}^N a_nt^n \in K[t]$ the polynomial truncation modulo $t^{N+1}$, and, following Bombieri's 1981 paper on $G$-functions, define the height of $f$ to be $h(F) := \limsup_N \frac{1}{N}h(F_{/N})$ as $N \to +\infty$.
Note that if a power series $F$ has positive radius of convergence at every place and $S$-integral coefficients for a finite set $S$ of places, then it has finite height. Thus the following claim settles the first question affirmatively under the dominant pole assumption.
Claim. If the Hadamard quotient $h := \sum_{n \geq 0} \frac{a_n}{b_n}t^n \in K[[t]]$ has finite height, then it is algebraic.
Proof. The power series $f \in K[[t]]$, being algebraic, has in particular a positive radius of convergence $\rho_0 > 0$ at the place $v_0$. On the other hand, the strict inequalities $|\gamma_i|_{v_0} < 1$ allow us to write, by expanding in geometric series,
$$
\beta^n/b_n = u_n + r_n,
$$
where the power series $u := \sum_{n \geq 0} u_nt^n \in K(t) \cap K[[t]]$ is rational, while the radius of convergence $R_0'$ of the remainder $r := \sum_{n \geq 0} u_nt^n \in K[[t]]$ at the place $v_0$ can be as large as desired. Then the Hadamard product $z := f*r \in K[[t]]$ converges on the disc of radius $R_0 := \rho_0R_0'$, which in turn may be taken to be arbitrarily large.
Let $C/K$ be the regular projective model of the finite extension $K(t,f) / K(f)$. Thus $t \in K(C)$ is a non-constant rational function on the algebraic curve $C$, which by design has only simple zeros: $f \in K[[t]]$ implies $K(C) \subset K((t))$. The divisor of zeros $Z$ of $t$ is a finite reduced non-empty subscheme of $C$, and $t$ is a local parameter at every point in $|Z|$. We can view the Hadamard quotient $h$ as a formal germ in $\widehat{\mathcal{O}_{C,Z}}$, the formal completion of $C$ at $Z$.
It is clear from the explicit description of coefficients of rational functions as (confluent) power sums that the Hadamard product of $f$ with a power series in $K(t)$ is again in $K(C)$. Therefore $f * u \in K[[t]]$ is the expansion around $Z$ of a rational function $y \in K(C)$ on $C$, and we have expressed our Hadamard quotient $h$ as
$$
h(\beta t) = y + z, \quad y \in K(C), \, z \textrm{ convergent on } |t|_{v_0} < R_0.
$$
Let $D$ be the polar divisor of $t$, and consider the ample line bundle $L := \mathcal{O}(D)$ on $C$. It has a canonical global section $s_0 := 1_D$ with divisor $D$. By Riemann-Roch, there exists a $k < +\infty$ and a non-zero global section $s \in \Gamma(L^{\otimes k})$ such that the section $y\cdot s$ of $L^{\otimes k}$ is regular outside $D$. Letting
$$
U := \{ |t|_{v_0} < R_0 \} \subset C_{v_0}^{\mathrm{an}} \setminus |D|,
$$
we thus have $y \cdot s, \, z \cdot s \in \Gamma(L^{\otimes k}, U)$. I claim that the conditions $h(h(t)) < +\infty$ and $R_0 \gg 0$ imply that the formal germ $F(t) := h(\beta t) \cdot s/s_0^k = (y+z) \cdot s/s_0^k \in \widehat{\mathcal{O}_{C/Z}}$ is a polynomial in $t$, hence in particular a rational function on $C$ As $\beta \neq 0$ and $s/s_0^k \in K(C)$ is a non-zero rational function on $C$, this will prove the algebraicity of $h(t)$.
Arguing by contradiction, assume to the contrary that the Taylor series of $F(t) \in K[[t]]$ is infinite. Then there are arbitrarily large values of $N < +\infty$ for which $t^N$ appears in $F$ with a non-zero coefficients $e_N \in K \setminus \{0\}$. The product formula gives
$$
\sum_v \log{|e_N|_v} = 0.
$$
The sum of the contributions over any subset of the places is bounded by $h(e_N)$, and we have the upper bound
$$
-\log{|e_N|_{v_0}} = \sum_{v \neq v_0} \log{|e_N|_v} \leq h(e_N).
$$
An estimate of the height is obtained by applying the bound $h(\alpha_1+\cdots+\alpha_r) \leq \log{r} + \sum_v \max_j \log^+{|\alpha_j|_v}$ to the sum defining $e_N$ as the coefficient of $t^N$ in $(s/s_0^k) \cdot h(\beta t)$:
$$
h(e_N) \leq \log{N} + h(h_{/N}) + h(\beta) + h\big( (s/s_0^k)_{/N} \big).
$$
The assumption that $h(t)$ has finite height means that $h(h_{/N}) = O(N)$, while the Taylor series of $s/s_0^k$, being algebraic, has $S$-integral coefficients (Eisenstein's theorem), and therefore a finite height. We obtain:
$$
-\log{|e_N|_{v_0}} \leq A_0 \cdot N,
$$
with a constant $A_0 < +\infty$ independent of $N$.
I will now establish the reverse bound $-\log{|e_N|_{v_0}} \geq N \log{R_0} - O(1)$ as $N \to +\infty$. Since $\log{R_0}$ can be taken arbitrarily large, in particular larger than $A_0$, this will lead to the desired contradiction, proving the claim.
Consider the line bundle $L_{v_0}$ on the closed Riemann surface $C_{v_0}^{\mathrm{an}}$ induced from the pair $(C,L)$ under the complex embedding of $K$ corresponding to $v_0$. Choose any $C^{\infty}$ hermitian metric $\|\cdot\|$ in $L_{v_0}$; it induces a metric in all tensor powers $L_{v_0}^{\otimes n}$, which we will continue to denote by $\|\cdot\|$. As in the proof of Prop. 3.6 in Bost's paper Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems, consider any $C^{\infty}$ function $\psi : C_{v_0}^{\mathrm{an}} \setminus |D| \to \mathbb{R}$ such that $\psi|_Z = 0$ and
$$
\frac{\sqrt{-1}}{\pi} \partial \bar{\partial} \psi \geq k c_1(L) = c_1(L^{\otimes k})
$$
as a point-wise inequality of $(1,1)$-forms on the Riemann surface $S := C_{v_0}^{\mathrm{an}} \setminus |D|$, where $c_1(L) := \frac{\sqrt{-1}}{\pi} \partial\bar{\partial} \log{\|s_0\|} \in A^{1,1}(S)$ is the Chern form of $(L,\|\cdot\|)$.
For example, for any two holomorphic functions $h_1,h_2$ on $C \setminus |D|$ which vanish along $Z$ and have disjoint ramification divisors, the choice $\psi := A(|h_1|_{v_0}^2 + |h_2|_{v_0}^2)$ is admissible for a large enough constant $A \gg 0$. As $Z$ is reduced, such $h_1,h_2$ clearly exist.
Since $s'$ is a section of $L^{\otimes k}$, we have the Poincar\'e-Lelong distributional equation (really the Cauchy residue formula here)
$$
\frac{\sqrt{-1}}{\pi} \partial \bar{\partial} \log{\|s'\|} = \delta_{\mathrm{div}(s')} - c_1(L^{\otimes k})
$$
in the sense of currents on $S$. Since by construction the divisor of $s'$ on $U \subset S$ dominates $N \cdot Z$, we can retain from the sum of the last two displays the inequality
$$
\frac{\sqrt{-1}}{\pi} \partial \bar{\partial} \big( \log{\|s'\|} + \psi \big) \geq N \delta_Z = N\frac{\sqrt{-1}}{\pi} \partial \bar{\partial} \log{|t|_{v_0}}
$$
of $(1,1)$-currents on $U$.
This means that the function
$$
\log{\|s'\|} + \psi + N\log{|R_0/t|_{v_0}} = \log{\frac{\|s'\|}{|t|_{v_0}^N}} + \psi + N\log{R_0}
$$
from $S$ to $\{-\infty\} \cup \mathbb{R}$ is subharmonic on $U = \{ |t|_{v_0} < R_0 \}$. Since $\log{|R_0/t|_{v_0}} = 0$ on the boundary $\partial U$, the value of this subharmonic function is bounded on $\partial U$ by $B_1 := \sup_{\partial U} \big( \log{\|s'\|} + \psi \big) < +\infty$, and the maximum principle implies that this last quantity bounds the value at a point $P \in |Z|$ (which we choose and fix). Since $\psi|_Z = 0$ by construction, the conclusion reads:
$$
\lim_{p \to P} \log \frac{\|s'(p)\|}{|t(p)|^N} \leq -N\log{R_0} + B_1.
$$
Since $e_N$ is the coefficient of $t^N$ in the expansion of $s'/s_0^k$, the left-hand side is at least $\log{|e_N|_{v_0}} - B_2$, for a certain constant $B_2 < +\infty$ which depends on $k$ and on the choice of metric $\|\cdot\|$, but not on $N$. Thus, with $B_3 := B_1 + B_2$, we get the required majorization
$$
\log{|e_N|_{v_0}} \leq -N\log{R_0} + B_3,
$$
completing the proof of the claim.
