Those "rice" type things are a sign of tapeworm. They get tapeworm from eating fleas infected with the larvea. Your other cat won't get them unless you have fleas now, and the other cat eats an infected flea.
I would call that a very bad infestation of tape worms. I'm sure the stray has fleas galore and you should make sure to change clothes and weash whatever you are wearing immediatly to keep the fleas out of your house.
This bad of an infestation could kill your stray and it must be seen by a vet asap!
No fleas. I combed him and got none. But he and my two have Advantage on them. He appears to have been well cared for and wandered from home, I went door to door to 60 homes and left fliers, and nothing happened! Our mobile homepark is a popular cat dump site! Anyway he had a vet appt. today for shots and worms and bloodtests. Linda