1. May one do oral surgery on Friday when he knows he will have to take medications on Shabbat and be in pain through the Shabbat?
There is a known machloket between the Ashkenazi halacha and Sefaradi regarding performing a delayed Brit Milah before Shabbat, as it will cause the person to violate the Shabbat (משנ״ב סי’ שלא ס״ק ל״ג). The Ashkenazim are lenient as the Mishna berura wrote, and the Sefaradim go with the stringent psak of the Chida (יו״ד סי’ רסב סק״ד) and Ben Ish Chai (רב פעלים ח״ד יו״ד סי כח).
At first glance, one can compare this to our question of surgery, which will cause him to take medication. However the truth is that everyone will allow it in this case, as taking medication is only Rabbinically prohibited and especially if the person started taking them before Shabbat.
There is another halacha in Shulchan Aruch (סימן רמח) which discusses taking a boat ride or going on a cruise which will cause the person to violate the Shabbat with various Melachot. The Ba’al Hamaor, which the Shulchan Aruch brings lehalacha, says one isn’t allowed to start something from Wednesday onward, if it will make him violate the Shabbat. Therefore doing an operation which isn’t urgent and can easily be changed to a different day should be done, and rescheduled to a day before Wednesday if it will require him to violate the Shabbat. Also here we can be lenient since doing the oral surgery doesn’t require any Shabbat violation.
Regardless, it is better to pick a different day for this procedure since doing it on Friday will most likely cause him to be in pain and prevent him from the mitzvah of Oneg Shabbat. (חוט שני פ״ח אות א’)
2. As winter is approaching, my lips start chapping and getting cuts. May I use chap stick or cream to soothe it?
Using such things are a Shabbat violation of smearing ממרח , and must be avoided. It is also prohibited if the lips are already chapped or dried as it is Refuah, part of the general decree not to use medicine on Shabbat or anything which heals.
But if it’s used as a preventative one may use liquid cream which he can prepare before Shabbat by adding water to the cream and with that the prohibition of smearing doesn’t apply.
3. Does fish still have a requirement to be cooked by a Jew in order to comply with the laws of Bishul YIsrael or since fish is eaten raw in Shushi we can be lenient?
Indeed, in order for a cooked item to be prohibited, it must be such that people don’t mostly eat raw. However, being that most fish is still cooked prior to consumption, this wouldn’t apply here and the laws of Bishul Yisrael therefore still apply.