First time cat owner (is my cat acting strange)

okay so last week on easter Friday a cat basically appeared on my brothers doorstep and invited itself in, he can't keep him because his dog gets very territorial and he doesn't want to leave the cat alone with the dog so I volunteered to take the cat.

anyway the cat clearly isn't a stray since he knows how to use a Litter Box and isn't shy at all he hasn't tried to scratch or bite me, my brother or the dog in the last 4 days. He does wanna go outside a lot which I don't really like because I'd want to keep him as an indoor cat if he stays with me but he isn't attempting to run away or leave in any way, he's also acting very affectionate and trusting he's always purring, making biscuits and sleeping on top of both of us and nuzzling in our faces.

We'll do a complete vet checking and check if he's been chipped or not today but honestly I'm just a little confused how the cat who had matted furr and over 10 ticks and concerningly thin just 4 days ago is so trusting and affectionate off the get go, the only cat I ever had before died when I was 5 or 6 and the few cats my grandparents had over the years were outside cats who wouldn't really come near us for more than 5 minutes so I'm just confused about weather this is normal or if there could be some underlying issue or something to be concerned about, like don't get me wrong it makes me happy that the cat is so comfortable around us but what if I do let him go outside and he just goes to the next family that let's him inside and offers him cuddles or what if it's connected to a medical problem that we're not aware of yet I'm just a little confused about the situation