#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?
Yes and Yes. In fact you should determine if the two circuits are a muntiwire circuit or if there are two neutrals going back to the panel. If it is 2 neutrals, split them and the loads. If it is a genuine multiwire, be sure it is marked and properly phased, then install a GFCI receptacle on the load side of the split.
Re: GFI kitchen replacement receptacle
Technically a GFCI does not work off of a ground wire at all. One is not even needed for it to function correctly. The functioning of a GFCI is as follows: the current going into the GFCI must be matched by the current leaving the GFCI by the nuetral (grounded)wire. If there is a difference between 4 to 6 milliamps then the GFCI will trip. This is why many 2 prong recepts are replaced with a GFCI just for safety and not for the equipment ground. The difference is telling the GFCI that amperage is missing and going somewhere it should not be going to.
As far as using a GFCI breaker for the kitchen that is not necessary and expensive. Replace the 2prong recept with a GFCI recpt and place the label provided with the GFCI "GFCI Protected" "No equipment ground"