I don't think that there are any jazz clubs in Rogers Park, but The Green Mill is a short el ride away (if you're near the el, of course). It's also the quasi-birthplace of the poetry slam, if you're into that sort of thing. The redline (el) runs all night, but the trains get spread out when it's late. Most of the busses don't run at night.
It looks like the best way to get near the Sears Tower from Rogers Park is the Metra Union Pacific North Line. You can map out public transportation trips here:
http://tripsweb.rtachicago.com/. I'm a big fan of commuting to work on a train, because you can read and/or listen to you ipod or whatever. It's much more relaxing than driving. Of course, the main transportation concern without a car is how to get to work and how to get groceries. There are some good independent grocery stores, for example, Morse Market and The Devon Market have cheap produce. For more traditional American and/or processed foods, you can look up the websites for Jewel and Dominicks--the two big chains in the Chicago area.
Rogers Park is touted as the third-safest neighborhood in Chicago. Make of that what you will. There is high crime in certain spots. If you see a big blue light on a telephone pole, that marks a police security camera and likely indicates that area is a hub for crime. I never had an apartment broken into and I never felt my family's safety was threatened inside the apartment. The place I lived with my wife and stepdaughter was on the fourth floor--so I figured it wasn't low-hanging fruit if someone was determined to rob our building. I never heard of any theft in anyone else's apartment (in the complexes I lived in), either. A bike was stolen from a storage space at my mom's old place in Edgewater, and the coin boxes on the laundry machines were pryed out at one placed I lived at, but that's really not the same thing. The only crime ever committed against me was having my car window knocked out when I lived in a not as densely populated part of RP. As far as I know, it was part of a string of vandalism and nothing was stolen. You can check the crime states for a specific address, intersection, ward, zip code, or whatever here:
http://www.chicagocrime.org/. There were a few homicides in the neighborhood when I lived there. Like I said, the relative safety varies a lot from block to block. My mom and her husband live in a pretty nice part near Loyola University and I'm not in the least bit concerned for their safety. My mom isn't out walking late at night, though, so take that into consideration. For a very pessimistic look at Rogers Park crime and politics, you can look here:
http://morsehellhole.blogspot.com/If/when you get a car, parking will probably be a bitch. The closer to the lake and redline (both on the east end of Rogers Park) you get, the denser the population and thus fewer parking spots. You might want to get a place with private parking when that happens. Alternatively, there are streets with permit-only parking at certain hours, and it's easier to find a place there if you buy a permit. We generally would not be out with the car past a certain time in the evening so that we wouldn't have to park 1/4 or 1/2 mile away from our apartment. The city takes advantage of the parking crisis with archane laws so they can bring in lots of revenue. Rather than setting a standard schedule, they put up temporary signs indicating street cleaning will happen on a given side of the street in two days or so. If you parked far away from your apartment, didn't use your car for a few days and those signs went up, you'd likely be up shit creek. Also, it's inevitable that your car will get nicked up from people parking in a tight spot behind or in front of it.
Most apartments will be old--often with nice hardwood floors and wood trim. I could turn off the heat in all of the places I lived in--one with a wall-mounted thermostat, one with a knob on the radiator, and one only with pliers. Most places don't have central a/c, so you'll have to get a window unit--especially if you live on a higher floor. Rent is pretty affordable compared to other northside neighborhoods. We negotiated our rent down to $855 for a two bedroom unit.
Now, the good stuff. Rogers Park is the most diverse neighborhood in Chicago, and supposedly one of the most diverse neighborhoods in the country. This affects you how? Well, for one thing, a variety of ethnic foods offered in grocery stores. Next, great restaurants. Lots of Indian places on Devon, the Ethiopian Diamond (my favorite place) neaby in Edgewater, taquerias, J.B. Alberto's pizza, etc. As far as open space, there is the lake on the east side (totally makes up for not having a yard if you take advantage of it) and Warren Park on the west side.
I don't know much about Logan Square other than that I know people who live there and like it. You could probably get to work faster if you lived there. Let me know if you're coming to town again and I'll see if I can show you around.