Can i leave for vacation and get back to US again with no problem?

E2 permits one to travel freely in and out of the US.

If I am illegal in US-can I apply for student visa being in US? Thank you

Going to be difficult to prove intent being here. It requires proving ties to your country among other things.

I-94 for asylee

It is 11 digits and in a row of boxes up on top left side.

Why did I get a different SSN?

Best for you to visit with your local SS office.

When applying for adjustment of status, will the USCIS know that my name was searched in the NCIC?

Only arrests or convictions matter in immigration processing. Not searching records.

Can you adjust status if you owe money to the IRS even though you are paying the IRS scheduled installments?

Your IRS situation as explained does not affect your I 485.

You do not say how your fiancée got to US or if you are a citizen or what and that matters. Second, this section does not deal with family law but...

Do i have to wait till my case gets approved or denied to apply for green card for marriege?

Provided your girl friend is a US citizen, you can apply after you are married.

If I-485 application was mailed to but returned by USCIS (due to Oct VB revision) does it constitute a "filing" of the I-485?

Technically you have filed form I-485 even though it was rejected.

I have a question about Visa Application for K1 visa

Well the problem is you have tipped your hand that he is not coming to visit but get married. On that basis probably better to stay with K1