feelings, emotions and facial expressions

1.Are feelings, emotions and facial expressions universal across borders, happy, sad, etc..?

Of course they are.
It is universal across borders that how people feel, althogh it can be different how much they expresses by the countries.

2. Try to name as many feelings as possible in Japanese.
Then, write the English translation.

喜ぶ　 悲しい　怒る 　悔しい　 寂しい　楽しい　後悔する　 心配する　怖い　うんざりする...
happy sad　 angry 　resentful　 lonely　joyful regretful 　 anxious 　scare 　boring...

3. Are there any feelings in Japanese that cannot be translated into English?

Yes, there are.
It couldn`t be traslated exactly that words which mean different level of feeling are.
For example, if I want to explain about おこる and いかる、
both words can be translated as `angry`, and It was difficult to find the words which have exactly same mean.