Tapeworm passed whole

I'm caring for a friend's cat for a couple of weeks. I've noticed the
telltale "rice grains" around the cat's backside before, but didn't
know what it was. Clearly my friend didn't know either or else I
wouldn't have to post tis message. Anyway, last night, after using the
litterbox there was a long (around 6 inches) string hanging from the
cat's anus -- turns out it had begun passing a tapeworm and the length
of it was just dangling there, exposed. After an emergency midnight
run to the vet, he's been given de-worming medication.

A couple of questions for the experts here...
1) Has anybody else encountered this? My wife and I were seriously
freaked out when we saw this thing. We honestly didn't know if it was
a piece of string, a worm, or part of the poor beast's intestine coming
out.

2) How long does the anti-tapeworm medication typically take to destroy
tapeworms living in the digestive system? Is it quick enough that
there is little chance of this happening again?

3) Is there any chance my cat could be infected? They've only been
together a couple of days, my cat *never* goes outside, and my friend's
cat has no sign of fleas or any other parasite carrier.

Thanks in advance for your help!