Minpins are tiny too, and my big Boo and Jemma are both terrified of the doggy next door. When they are on the balcony and he gets let out, they RUN inside.
A dominant little dog does take his place in the home really quickly. Yes, big dogs can be a danger to little dogs, but how will you know how your dog will react with a little terror unless you try (or if you have met some before).
And the yard thing? I totally disagree with a yard. Dogs don't want a yard, they want any space anywhere as long as YOU are in it. So many people with yards either leave them out alone or leave them out tied up alone. Dogs don't want that at all. I'd much rather adopt to somebody who lives near a park than someone with a yard. Without a yard you have to take an active interest in your dog's daily outings.
Our dogs are never outside without us-- we have a tiny apartment in the second floor. We have a balcony that they can watch squirrels from, but even then, they don't stay out for more than a minute or two when we're not there or the door is shut. We would probably NEVER be accepted for adopting a dog from a rescue in this apartment. We got Boo from the SPCA on a weekend (not so knowledgeable staff on weekends and they don't screen anyway). But Jemma we got from a lady who was moving. She wanted a person with no yard. She wanted her dog to have as much one on one time as possible.
I would be more worried, if I was a rescue, about people with a fenced in yard than without-- people just use yards as dog-sitters.
Anyway, I'm sorry you got turned down, but maybe there is a person who has to give up their doggy somewhere in your area that will find you.