@roshke, the connection might be your business if that person listed it on their about page. Or it could simply be that person is FB friends with some FB friends of yours. I get lots of suggestions for people who are friends of friends.
I don’t believe I see everyone’s likes in my newsfeed. I sometimes see a friend’s like of a public post such as buzzfeed or a news organization. But I don’t see when a friend likes a personal comment from a friend of theirs.
I volunteer at two local businesses, and will regularly share their FB postings with my friends. One is a garden, the other is a museum. I would hope those postings are not an annoyance to my friends.
I frequently “like” a personal posting from a friend. It’s a way to let that friend know you saw it without commenting on it.
I will see when some friends of mine have wished a mutual friend a happy birthday, but only if I haven’t wished that friend a happy birthday yet. I’ve never noticed it if I post birthday wishes first thing in the morning. I have no problem if a mutual friend reads my birthday greeting.