This sounds like retaliatory eviction which is a defense to an eviction proceeding, but it will get you only another year in the apartment. This...

Will I get in trouble for not paying rent after getting an eviction notice?

Come to court prepared to explain everything. If this is a holdover proceeding, you will have some time to move, but not all that much. You should...

Pending Eviction

This is part of the game. Your lawyer should carefully document what is going on, but do not be surprised if this happens a few more times.

The court will probably not allow this amendment. You should consider starting from scratch.

What are the chances that the building dept in NY will seek a warrant for a basement apartment ?

You can expect to have a very expensive violation issued for this apartment.

Landlord let contract expire.

That lease no longer binds you. You need only give one month's notice to leave.

You can certainly sue, but the chances of your actually recovering something are very small.

Can a tenant state they are suing you for one thing in their claim and then add more to it during court?

You can take an appeal on this, but you should not expect to win and it can be expensive and you don't get reimbursed for your expenses.

Can you get your Spouse off your NYCHA lease without him being present ?

Discuss this with your building manager who will give you guidance.

Landlord and tenant lease?

Generally speaking you cannot be bound by somebody else's agreement, but you should have an attorney look into this.