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

I think we have a lot of universal across borders.
But it is a certainly that only human have a "emotion".
Therefore we have a lot of common emotion.

2. Try to name as many feelings as possible in Japanese. ( 10 minimum) Then, write the English translation

There are a lot of feeling words in the world.
Tell you a few example.

楽しい　　(happy)　　　
嬉しい　　(glad)
面白い　　(interesting)
腹が立つ　　(angry)
悲しい 　(sad)
心配する（anxious ）
妬む　(envy)
疲れる　　(tired)
恥ずかしい　　(ashamed)
退屈　　(boring)
緊張する　　(nervous)

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

Yes,a lot!!
Japanese has a lot of humility word.
For example,
If you are want to say "今後ともよろしくお願いします"
What are you going to say?
Will you say "Nice to meet you" in English? Is this same meening?? NO!!!
I think it is simmilar meening, but diffarent, Isn't it?

It is very difficult for Japanese to tell humble words for foreigner.