OK, I probably missed some stuff as I scrolled, but I’m writing this to reply to a few vocabulary/grammar points I spotted in the translations that have been done recently.
First of all, in this context, 様 is probably read さま, because it comes after a person’s name/title. お父様 (otou-sama) is just a block that means ‘your father’. A term of address like ‘san’ or ‘sama’ has to be added to the word ‘father’ because of politeness: the professor is referring to someone else’s father, not his own.
なくなりました is a verb of sorts, but it’s really just ない (adjective/verb meaning ‘does not exist’) being turned into なく (its adverbial form) and being added to なりました (to become). So it really just means ‘became non-existent’ or ‘has become non-existent’. So I would have translated it as ‘Your father’s hope of recovery is almost gone/has almost disappeared/has become almost non-existent.’ (Of course, the most natural way to say it would be ‘Your father’s hope for recovery has dropped to close to zero.’ However, that’s quite different from the Japanese sentence.) Either way though, the translation as a whole was quite accurate, which is good.
I’m honestly not sure, but I’d say probably not. The ‘somehow’ indicates that he felt it was unlikely, which matches the fact that he thinks the chances of recovery are very low. In an anime, I once saw an exchange like this: ‘Are you ok?’ 「ええ、なんとか」The guy who said なんとか was a very humble, pleasant noble, and he was out of breath after running from a monster.
This is correct, and yes, it is more formal. Or closer to written language. Ichidan verbs are the ones that end in る and for which る vanishes when you add ます. Godan verbs are the ones for whom the ‘u’ sound becomes an ‘i’ sound when ます is added. Ichidan = the verb stem only has one form which doesn’t change. Godan = the verb stem changes depending on whether it’s negative, imperative, linked to ます etc. There are five forms in total. In essence, the usage of 加え you see has the same meaning as 加えて, but it’s formed by taking the ます stem of the verb (you take the formal ます form and remove the masu to get this).
Bonus (but I don’t want to overload everyone with information, so ignore this if it’s confusing): in formal written language, the masu stem is used to end clauses, while the て-form links verbs within clauses. I can’t think of a good example now (my fluency has limits, as you can see), so please just take my word for it. You can find examples in NHK news articles when there are long sentences. The simplified way of understanding it: the て-form shows a slightly closer link between verbs than the ます stem in the same position.
Yes, you’re right. It’s a grammar point in this case: 〜てやる means ‘to do 〜 for someone else’. It’s similar to 〜てあげる.