Is everyone obligated to follow the fatwas of his country, and if we adopt the fatwas of scholars of other countries, is it considered a sin?

All perfect praise be to Allah, The Lord of The Wolrds and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

It is permissible for the person who isn't specialized in Sharia sciences to apply the opinions of renowned scholars whom he trusts ,whether they be from his own country or not ,but if the scholars have different opnions regarding a certain issue/matter, then he must consult someone who is more knowledgeable than himself .It is preferable that you (the questioner) specify the case of your interest ,so that we could give you a more specific answer since some scholars deliver fatwas based on illogical/atypical opinions which should not be applied no matter what .And Allah The Exalted Knows Best.

Is it permissible to offer an absentee funeral prayer?

Yes, it is permissible to offer an absentee funeral prayer.

Is it permissible for the doctor to uncover the hair of a female patient out of necessity?

It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams(unmarriageable kin).

I work in a government institution that gives an additional three dinars per hour I work. The manager combines students into a single session because attendance is low, and the material taught to all groups is the same. I am paid for two hours, although I actually taught only one session, noting that my transportation costs one and a half dinars. What is the ruling?

The only amount you are permitted to take/have is for one hour work. And Allah Knows Best.