Q. Can I accept reimbursement for costs of flight?
A. Generally no, unless both organization has exemption and you and organization fully comply with its requirements. Organizations with exemption currently are :CAP, EAA, AFMA, etc.
Further explanation and Links to regs

Q. Can I take tax deduction?
A. Yes. FAA considers it to be compensation, but has repeatedly stated policy . . .
Further explanation, links to policy statement and existing content.

Q. Can I use Compassion (CMF) call sign?
A. Yes, if you are flying patient . . .
Link to existing content

Q. Can I bring passenger’s oxygen bottle on board?
A. Generally no. May use oxygen on board, or pilot may supply oxygen, some O2 concentrators permitted.
Links to FARS, SFAR