1. I was planning on opening a Trad IRA to lower our tax liability but TurboTax stated that we make too much (>115K). i thought the limit was >178K or in that area for claiming a full/partial deduction?
$178K is the limit for making direct contributions to a Roth IRA; you are below that limit, so you can contribute to a Roth IRA for 2013, but it won't lower your taxes.
$115K is the limit for making deductible contributions to a Traditional IRA if you are covered by an employer plan. (If one of you is not covered, he or she can make a deductible contribution but the other spouse cannot.)
2. What, if anything, can i do to lower my tax liability? (I'm not trying to dodge paying taxes. they are inevitable. just trying to see what i can do to lower it)
You can't do anything now; you owed these taxes as soon as you earned the money. You can tell your employer how much to withhold to pay taxes in advance, but you must pay the difference in April, and if you don't have enough withheld, you may owe a penalty. You can avoid having a big tax bill next year if you file W-4 forms with your employer with fewer withholding allowances.
3. We donated a bunch of stuff in 2013 while moving out of our apartment to goodwill, but don't think we got receipts. Any way we can claim deductions for this without having a receipt(s)?
You need a record of your charitable donations, but given the information you reported, even deducting it wouldn't help you; your itemized deductions are less than the standard deduction. (And this is part of the reason for your unexpectedly high tax bill; you aren't getting a tax benefit from deducting the mortgage interest, because you don't have other deductions such as state income tax.)
4. Does it matter if we file as married- jointly vs married separated?
You'll have to check it out, but it is likely that you will pay more filing separately.