Can i file for H1b visa in April,2016 while on the grace period of my J1 Visa. 2 yr rule doesn't apply in my situation.

I believe your grace period would need to bridge until October 1 because that is when the H-1B status would kick in.

Does the presence of mold impact the legal habitability of my apartment? If so, what are my landlord's responsibilities?

You should absolutely test the mold ASAP.

Does it require a specified time of residency? what is the shortest time of period for lease agreement that I could get?

California is usually very understanding of undocumented foreign nationals - look up their DMV regulations online.

What is the best legal way to bring my wife in United Sates. I am green card holder and my wife has a visiting visa.

You can file an I-130 Petition for her.

Am I Eligible for Employment-Based Greencard

Your company does decide what the requirements are for the job position.

Now is there a way we can get her I-94 extended from here or she will be out of status after Dec end ?

She can travel by herself as long as she has a copy of your I-797 approval.

How can I deal with this situation? how can I apply for an LCA and an extension of my H1-B

Your employer will find out what the prevailing wage is before filing the LCA.

Is it legal for a landlord to include mandatory carpet cleaning in the lease?

That is pretty standard lease language.

New York specifically is having issues with AirBNB.

Is the landlord at fault? If so, should I pursue this further?

What does your lease say about injuries on the premises?