"I will see you on Friday!", Yes "I will see you Friday", Yes (informal) "I went to the cinema this morning" yes, (added "the" ) "I went to the cinema in this morning" No. "I will go to the cinema in the morning" yes (future tense/aspect) "I will go to the cinema tomorrow morning" yes (future tense/aspect)

I went to THE cinema this morning. Never use "in".

"I will see you on Friday!" and "I will see you Friday" are both correct. "I went to cinema this morning" is incorrect. "I went to THE cinema this morning" is correct. "I went to cinema in this morning" is not correct. Change it to: "I went to THE cinema in THE morning" for perfection!

Yes, you can say either!

If you are speaking with Americans, it is more natural to say "I went to the MOVIES."