Nobody here is giving any "legal advice", free or otherwise...
Short answer: that would be illegal, not to mention unethical.
The laws for what you're describing are pretty clear: the person who created the original work owns the copyright, which includes the exclusive right to make copies, publish or distribute them, adapt them, or to display or perform them in public. Copyright is free and automatic and does not require © markings or registration.
That said, the applicant owns the copyright of the art he or she sent to you. If you were to "adapt" it from a sketch into a 3-D mascot, that would be a copyright infringement.
They would only "own" the parts that they created, so someone else would still own the copyright of the compilation of images you provided, as well as copyright on each individual image.
The basic rule on any copy that was given to you (of anything copyrighted) is that your possession of the copy has NOTHING to do with ownership of the copyright. Even if it is a unique artwork, like a stone sculpture, your acquisition of the only existing "copy" (chiseled in stone) would give you no particular right to have a copy made, because the original artist still owns the copyright on the creative work.
Yes. But I'd think they may want to put the child's name on the title, just to show they aren't converting it.