You are, but it can be touchy. It is rarer in US English than it is in British English and I would more likely hear a parent talk about "my Jenny" when they're talking about their child. I may also use "our" (not "my," but "our") if I'm talking about a person in a group (company, school, club) that we all belong to. But as Kate says, using it can have a nasty twinge of ownership. And you don't want to do that.

It's not uncommon in British English (and I've heard it elsewhere, too) to refer to "my Derek", "her Bob" or "our Lisa." "That <name>" as you have it means you're expressing a mild disbelief at something about the person. It would usually (but not necessarily) be negative.

No, you are never allowed to use possessive pronouns with a person's name. Because it implies ownership, the only exception to this that I can think of would be if you owned a slave (Lisa) and you were speaking to someone about her: "Stay away from my Lisa." There is not the same connotation with MY + nouns that are not proper nouns.Therefore, "my boyfriend" is perfectly normal and good thing to say.