If I'm set to Strict Cancellations for boarding, will I show up in search results for people who need a sitter on short notice?

Yes, you still will show up for clients who need a last minute sitter. I would imagine that if they are needing a last minute sitter chances of them canceling is not that high. Although, I am not positive on how it would work if they choose to cancel, But in the event that they do I believe it will then become an option to you.

For me personally have my cancellation policy scheduled as strict.

If a client did book a couple days before the stay started but than had to cancel, I personally would base my option to give them a full refund on if I had to turn away any dogs. if I did not, then I think it would only be fair to refund them the full amount considering that they did just book.