(の)かな is hard to translate into an exact English phrase. It’s used to express nuances like “I wonder if…”, but when speaking to someone it can also have the added nuance of asking the other person’s opinion as well.
Like in your case, “I wonder if the dog misses his previous owner, what do you think?” would be the nuance going on.
It’s like you’re making a guess about something and wondering if it’s true, but also looking for confirmation about it at the same time.
In Japanese it’s considered fairly direct to talk about someone else… well directly, so when you see or hear (person)のこと, it’s essentially just a way to soften the sentence. I guess you can think of it as “about (person)”. Literally 小林さんのことを思い出す means something like “remembering things about kobayashi-san” but the meaning is more or less the same as if you dropped the こと. It’s just to make it more polite or kind towards that person essentially.
誰かのこと means things about said person. “the way they look, talk, etc”, and it is indirect this way.
For the かな thing, か is doing its normal job but な is used to talk to oneself. It can also be used informally by males as a more “pushy” version of ね