#1 Replacing a non-grounding receptacle in a kitchen? Would a GFI work? Use a GFI circuit breaker for the circuit?

question #2 (complete unrelated to #1 different house with ground wire) - Replaced a receptacle that shared a box with a lights switch. The GFI tripped immediately. It turns out the light is on a different circuit. The ground wires were all tied togther in the box. Would this by itself trip the GFI?