What type of lawyer would I need for mold sickness day due to landlord neglect?

Most likely a personal injury lawyer would be a good place to start,

Given this information is this contract in force and if it is not will I be able to get my security deposit back?

What sort of housing code do you think needs to be on there? Seems easiest just to give the 30 day notice and leave.

Do I have any recourse? Is there anyone I can call about this?

Has there been any follow up with the main office?

You should be able to acquire your own insurance but most ESA are some sort of retriever and don't conflict.

What obligations does an establishment have to secure a tenant's safety? What constitutes an unsafe living environment?

Withholding rent is called self-help and is prohibited.

Is it legal for the prospective landlords to have cashed the first month's rent check?Is all this normal? What can/should we do?

It does seem like the landlord has accepted the offer - check with your realtor ASAP.

Is his asylum case can effect me if I am going back to my country to apply for F1 visa?

Please refer to other posting for additional information.

Do I need to apply for a waiver for my J-1 or can I just get married and let my husband fill out I-130 and me the I-485

They aren't asking about your vise they are asking about whether you are subject to 212(e) - there should be a notation on your visa about it.

Is his asylum case can effect me if I am going back to my country to apply for F1 visa?

His case should not affect you but yours may affect him because the asylum office may think he has another country he can safely live in.

Selected as the best answer

My H1B Extension is expired on 3rd December, 2015 but I applied my extension with employer on September, 2015

You should be ok but USCIS will be issuing new rules soon.