The first question can be answered by taking the idea of definable well-orderings which are not set-like. That is, we can consider the formula $\varphi(x,y)$ which states that $x,y$ are distinct ordinals and either $y=0$ or $x\neq 0$ and $x\in y$. It is not hard to see that $\varphi$ defines a well-order of order type $\sf Ord+1$ on the ordinals. This can go on, and we can ask (e.g. What is $\omega_1^{CK}(\mathsf{Ord})$?):

Given $M$ a transitive model of $\sf ZFC$, what is the least ordinal $\alpha$ such that $\alpha$ is not a first-order definable over $M$?

To the second question, the answer is in fact the the following things are equivalent:

1. The axiom of global choice.
2. Every class can be well-ordered.
3. There exists a well-ordering of $V$.

The implications $(3)\implies(2)\implies(1)$ is trivial. The proof that $(1)\implies(3)$ is about the same as the proof that the existence of a choice function on $\mathcal P(X)\setminus\{\varnothing\}$ implies that $X$ can be well-ordered.

Finally, the well-ordering of $L$ is of order type $\sf Ord$. To see this, simply note that if $x\in L_\alpha$ then $x$ does not appear before $L_\alpha$ in the order. Therefore we have a cofinal class of sets, which have only set-many predecessors. This is enough to show that there is no point which has class-many predecessors so the only order type fitting this is $\sf Ord$ itself.

The first question can be answered by taking the idea of definable well-orderings which are not set-like. That is, we can consider the formula $\varphi(x,y)$ which states that $x,y$ are distinct ordinals and either $y=0$ or $x\neq 0$ and $x\in y$. It is not hard to see that $\varphi$ defines a well-order of order type $\sf Ord+1$ on the ordinals. This can go on, and we can ask (e.g. What is $\omega_1^{CK}(\mathsf{Ord})$?):

Given $M$ a transitive model of $\sf ZFC$, what is the least ordinal $\alpha$ such that $\alpha$ is not a first-order definable over $M$?

To the second question, the answer is in fact the the following things are equivalent:

1. The axiom of global choice.
2. Every class can be well-ordered.
3. There exists a well-ordering of $V$.

The implications $(3)\implies(2)\implies(1)$ is trivial. The proof that $(1)\implies(3)$ is about the same as the proof that the existence of a choice function on $\mathcal P(X)\setminus\{\varnothing\}$ implies that $X$ can be well-ordered.

Finally, the well-ordering of $L$ is of order type $\sf Ord$. To see this, simply note that if $x\in L_\alpha$ then $x$ does not appear before $L_\alpha$ in the order. Therefore we have a cofinal class of sets, which have only set-many predecessors. This is enough to show that there is no point which has class-many predecessors so the only order type fitting this is $\sf Ord$ itself.