To improve relationships with strays, don't forget the bonus of some LTR's, namely Attractive (it gives a starting boost for any new Sims/Pets your Sim meets for the first time) and Never Dull (you can repeat the same interaction over and over). Never Dull is less useful if your Sim is Charismatic, because the "repeat over" feature is included in the trait.
I'm currently testing Riding skill as supermax skill for a Dynasty and here is what I found:
My test founder is Equestrian (but not charismatic) and has the 2 rewards. When she meets a wild horse for the first time, she reaches friend status after just watching the horse and letting sniff hands. Then she can adopt just adopt it. Super fast isn't it?
With dogs and cats, it's a bit slower because she is Equestrian and not Animal Lover, but she still gets a starting bonus that brings the relationship with the stray pet up to half way to friendship.
Then, I haven't' really tested this properly, but I think that when you invite a stray inside, the pet will tend so stay longer on your lot and come back more often.