Can the school I study in offer me a student stipend legally?

You should talk to your DSO but without some sort of employment authorization, this will not be possible on a F1 visa.

Is fear or phobia a mental disorder that may make someone inadmissible to enter USA?

Generally, you should not be inadmissible because you were institutionalized but you may be denied if you are likely to cause harm to yourself or...

Find out at the office which is adjudicating your application.

How can I make sponsorship without 20000 co sponsor?

Your fiance does not need a sponsor for citizenship. It's different if you mean LPR status.

Form N-400 03/26/16 N Edition - Should I answer yes to Question 27 if i ever been paroled ?

Yes because you were paroled into the U.S. upon return.

May I file I-130 and I-485 for my father?

No, since he did not enter the U.S. legally unless there is an exception for him.

Do i need to send original copy of my New I797 form to her or can she use photo copy. Her H4 is Stamp with my old employer.

A copy will be fine. Send it to her - she can use her H4 visa

Does green card holder need to sponsor spouce?

Yes, a green card holder needs to sponsor their spouse. Where is your spouse currently though?

Can I petition my minor child even though my green card expired last year?

Yes, though the evidence of your LPR status expired, your LPR status did not.

N-400 - After fingerprints in January 2016 - no response from USCIS - Can a lawyer help?

I'm not sure if a lawyer can "expedite this" but hiring an attorney might provide you with the knowledge and assistance to have your case...