I was terminated for "dishonesty" surrounding the use of benefits provided by my employer

There is not likely to be any recourse unless they were restricting your FMLA rights, which I don't see in your description.

Can owner tell renter to leave early from the contract because they want to sell the house in January of 2014?

Both the present owner and any future owner are bound to the lease with you.

I was just fired via email for missing one day of work to care for my sick 7-year old child. Is this legal?

Sadly, in this state, they are within their rights.

Is it illegal or unlawful to threaten to take additional action against someone if they won't settle?

That is a crummy way to litigate, but not illegal. At least, not yet. If she brings some baseless claims, then there may be something you can do....

You are still on the hook legally. Talk to the landlord (not the property management company). He/she may have a heart. If not, have a lawyer...

Does a panic attack count as an excusable neglect when filing motion to vacate?

Probably depends on the judge. It would help to have some sort of written confirmation from your doctor. Judges hear a lot of crazy excuses from...

Can they go straight to wage garnishment?

Once they get the judgment, they can garnish. So try to work out a deal before they get the judgment.

We rented in Apache Wells, upon getting here the place was occupied, we put a $1500 deposit down, how can we get it back?

Get a lawyer to write a letter to the landlord for you.

My downstairs neighbor keeps complaining about noise I'm not doing, and possibly getting evicted. Can i sue for harassment?

I suggest you start gathering statements from surrounding neighbors saying the sound is not coming from your apartment. Present them in writing to...

Can I keep a deposit when tenant breaks the lease

Probably. But you need to let a lawyer look at your lease.