beezneez mentions Hawthorne, it's a fun neighborhood. Belmont is close to Hawthorne, it's fine. Lloyd center isn't really a neighborhood, it's a commercial district but anything north of Broadway (just north of Lloyd center) close in is fine. Mississippi neighborhood is great in my opinion.
Mississippi is very close to PCC Cascade, which is the campus I'd choose to go to if I was to enroll in PCC. The other campuses are way out of town and have more young suburban kids who's parents are forcing them to go to community college. Cascade is diverse and the students are more motivated.
I only go downtown for a reason, I'd never choose to live there. I know people who like it but it's not hard to get from the close in east side to downtown.
I have a friend who does session work and knows some of the younger jazz guys and I'll ask him if he knows where they play. I am under the impression that Jimmy Mac's is the older jazz guys hangout but there is probably some kind of young innovative jazz scene going on somewhere in this town. I know I've waited on some of the younger jazz crowd at my bar and they were all super nice guys who I've head are excellent musicians.
There is a publicly funded jazz radio station in town KMHD 89.1 you might want to check it out. One of the DJ's come into the bar I work at from time to time and we've always hit it off. If I see him I'll get his number because I'm sure he is the kind of guy that wouldn't mind chatting with or going to a show with someone who just got into town. I don't see him all that often maybe once a month or so.