Can Someone Else Claim Your Mortgage Interest?

Do you mean like if you sold the house "subject to"?  Not sure what you mean.  Some people sell their houses "subject to" the existing mortgage meaning (roughly) that the new owner is taking over your existing mortgage.  You'll get the interest paid info in the mail but your buyer can take the deduction as long as you don't try to take it too.

If you mean you don't need the deduction for whatever reason (not itemizing or something) and someone else could use then, no, you won't be able to do that.

It depend upon who paid the mortgage.  Also, are you someone else's dependant?  Check with a tax person.  It doesn't have to be a CPA; in our town the senior center offers tax help.