Interesting. I've only had a short glimpse a Unger's paper. Does "$\mathbb P$ is absolutely $\delta$-c.c." mean that $\mathbb P \times \mathbb P$ is $\delta$-c.c.?

@JoelDavidHamkins Thanks for spotting that typo. I'll fix it.

Zetapology, that's actually not an issue. It's just that $\beth_{1}$ represents different ordinals in different models. If we fix the ordinal $\alpha$ - not the definition $\alpha$ may satisfy - as I did, everything works.

@Goldstern You're right, of course. I'm not entirely sure how to elegantly state what I have in mind but something like "for every $\alpha > 0$ there is some forcing $\mathbb P_{\alpha}$ such that $1 \Vdash_{\mathbb P_\alpha}^L \neg \mathrm{CH}(\aleph_{\check{\alpha}})$" would work. However, I don't think that focusing on this formality is very helpful to OP which is why, grudgingly, I consider keeping my statement as is.

So while $\mathrm{ZFC}$ proves that $\mathrm{CH}(\aleph_\alpha)$ occurs on a club class, it can't guarantee this for any particular $\alpha$.

Regarding 3: In this answer Andreas Blass shows that determinacy on length $\omega_1$ games over $\omega$ is inconsistent with $\operatorname{ZF}$.

Note that if we relax the phrasing of the first question a bit, we can get positive results: Let $U$ be a 'sufficienlty nice' measure on $\kappa$ (e.g. be derived from an ultrapower embedding) and let $i_U \colon L \to L$ be the restriction of the ultrapower embedding to $L$. This itself is an ultrapower embedding of the structure $(L; \in, U \cap L)$ (i.e. definable in this structure by the usual construction) and in there $i_U(\kappa)$ is weakly compact.

@Joel No, there is no such $f$. See Will's answer.

I see. Your kind of games look pretty different from the ones I had in mind. Thanks for your clarifications.