1) You must be legally obligated to pay it.
2) You must have actually paid it.
SO yes, if both of you meet the above criteria then each of you can claim what you actually paid. However, if your name is not the one shown on the 1098, if you are audited be prepared to prove you paid what you claim. Else the IRS will disallow the deduction and you'll pay a fine/penalty for claiming a deduction you didn't qualify to take. (If you can't prove it, then you don't qualify)
Yes. Since you are on the deed, you have "substantial interest" in the property and therefore meet the requirement to be legally obligated to pay the mortgage.
But, You pretty much have a choice. One can claim it all or you can split it. It's usually best if only one claims the deductible home expenses, allowing the other to use the standard deduction.