I can see why you choose to live with your dad
You dont need both signatures as already stated
Just because you turn 16 does not mean you can get an adult passport, you can still use the one you have until it expires, but of course your mother has it.
I fail to see how it is legally hers just because she paid for it. It is in your name, she cant use it. She is being awkward and obstructive - personally, I think it will be easier for you to report it as lost to the police and carry on from there.
How soon before you travel do you turn 16?
Oh, and dont forget to send her some photos of you laying on the beach in some exotic country ;)