Both questions are answered in the paper you refered to (i.e., "Recognizable sets and Woodin cardinals: computation beyond the constructible universe") https://arxiv.org/abs/1512.06101 in Lemma 3.13 (I will write "OTM-aw" and "OTM-ew" for "OTM-accidentally writable" and "OTM-eventually writable":
Concerning accidental writability: You are right, the OTM-aw real numbers are exactly the constructible ones. More or less, one can see this in the way you indicated by simultaneously simulating all OTM-programs in all parameters and, whenever one of these outputs a real number, one writes this to the output section.
Concerning eventual writability: If $\eta$ is minimal such that $L_{\eta}$ is a $\Sigma_{2}$-submodel of $L$ then $x\subseteq\omega$ is OTM-ew if and only if $x\in L_{\eta}$.
However, it is possible, as you indicate, to see that the supremum of the OTM-ew ordinals must be countable without determining its value: Each coded ordinal will be countable in $L$, and there are only countable many programs, hence at most countably many OTM-ew ordinals. Finally, the function $f:\omega\rightarrow\omega_{1}^{L}$ mapping $i$ to the ordinal eventually written by the $i$th OTM-program (if it exists) and to $0$, otherwise, is definable in $L$, hence contained in $L$, and now the supremum of the OTM-ew ordinals is $\bigcup f[\omega]$, which is countable in $L$.
I understand that part B is asking what is wrong with the following method to eventually write the supremum of the OTM-ew ordinals: Run all program simultaneously, write a code for the sum of the outputs to the output section. At some point, all programs that stabilize have stabilized and then, the output will be stable and equal the sum of all OTM-ew ordinals.
The problem is that, if we do this with all programs, then the output will not stabilize, because it will include the outputs of non-stabilizing programs. It would work if we could restrict ourselves to those programs that stabilize; however, the set of programs that stabilize is quite complicated and in particular not OTM-ew (basically, this section is a reductio proof of this), so this also does not work.
edited: Previously, this post stated that $\eta$ merely needs to be minimal such that $L_{\eta}$ and $L$ have the same $\Sigma_{2}$-theory, which would mean that it is an $L$-countable ordinal. As Joel Hamkins pointed out below, this is false.
If you read the sentence "just before" definition-3.10 then you will see the expression $x \subseteq \gamma$. So if $\gamma$ is some ordinal then it seems to me that the author(s) are talking about some subset $x$ of the $\gamma$. That is a set of ordinals whose where each individual ordinal must be less than $\gamma$. So to define "accidentally writeable reals" specifically, you just need to set $\gamma=\omega$ (in definition-3.10 for the paper you linked).
Further regarding this point:
But I want to ask about reals accidentally written on the initial segment of length $\omega$ (the smallest limit ordinal) of the output tape at some time $\tau$, assuming that $\tau$ is not necessarily countable. I will call such reals (or ordinals encoded by reals) “accidentally $\omega$-writable”.
It seems that no new accidentally writeable real can be generated at some time $\geq \omega_1$. That's because if it was possible, then given some specific real number as input to OTM, it would be possible to halt at some time $\geq \omega_1$, which is impossible [ for this, start enumerating all accidentally writeable reals and when a new accidentally written real matches the "input real" then halt ]. For this, I don't know the reason, but I think that the answer to one of your older questions is relevant in this regard https://mathoverflow.net/questions/372051. I am assuming that the answer given is correct (since I don't understand it).
Edit: Sorry I misremembered the reference I was talking about.
Regarding this point:
I have read that the accidentally writable reals are exactly the reals in $L$ (part 3 of Lemma 3.13 in the linked paper). Does this imply that the supremum of accidentally $\omega$-writable ordinals is equal to $ω_1$?
This should be equal to $\omega^L_1$. It is consistent with set theory that $\omega^L_1=\omega_1$. It is also consistent that $\omega^L_1 \neq \omega_1$. I don't know the reason, but the answer in this question seems to mention this: Existence of bijections in $L$.
Best Answer
As Andreas Lietz stated, the only difference between "$X\prec_1 Y$ and "$X\preccurlyeq_1Y$" is that the former requires $X\not=Y$ while the latter allows $X=Y$. So they don't behave differently in this situation (in particular, note that $L_\alpha\not=L$ for every $\alpha$ since $L_\alpha$ is a set but $L$ is a proper class).
The choice of one over the other just reflects a notational preference on the authors' part: do we use "$\prec_1$" since it conveys slightly more information "up front" or do we use "$\preccurlyeq_1$" since that extra information is really trivial anyways and the more flexible notation is generally nicer? (Personally, I lean on the side of "$\preccurlyeq_1$" in all situations where it works, but that's just me.)