Re: Keep getting stray cats in our house.
Welcome to Cat Forum and I understand the problem you have, unfortunately, there aren't many solutions. I am happy that you understand these stray cats are only trying to survive, but I also know ... there *is* a limit as to how much someone can give, as I faced that problem by feeding strays on our rural property. I was able to find an organization to help me TNR (Trap Neuter Release) and most of these adult cats eventually tamed and became our housecats. The kittens, I tamed, socialized and fostered in our home until they were ready to be spayed/neutered and go to the adoption center to find their own permanant homes. That solution worked for me.
I understand there is a limit to what any person can financially do. I had to find an organization to help me when I realized that my feeding would allow a higher survival rate, until the number of cats exceeded my ability to feed, then it would cycle through survival-of-the-fittest until the weaker cats had died off, food resources returned to being plenty for all and the cycle would begin again. My solution was to trap everyone I could for spay/neuter and tame/socialize for adoption all of the kittens.
We also had an adult cat who would not use the litterboxes. He was a former feral who I tamed outside our home before I began to let him inside. He eventually grew comfortable spending a lot of time indoors, and even overnight, but we had to let him in/out so he could go potty. He never had an accident inside, until he became ill, and that was understandable and easily forgiven.
In your case, the ONLY way to stop strange kitties from coming in your home is to deny them access.
Two solutions would be a cat-flap door with an electronic reader that recognizes a specific chip-thing that is placed on your cats' collars. I understand batteries and/or replacement chips are sometimes required and a little spendy to have to replace. Another would be to create an outdoor cat kennel that is attached to your home, giving your cats free-access outdoors and into a safe environment created by you. We plan on building one along the East side of our home. I am hoping we can make it 20'x45', but we may have to begin with 10'x10' at first, and then add on as finances allow. A final solution would be to trap the stray cats and have them removed. This usually will mean they would go to a shelter that may or may not work with taming them and helping them to find homes, and that could ultimately mean a death sentence for them.
There are few easy solutions, no easy answers and no guarantees. I wish we *could* save them all.
Best of luck with your situation.