I may have missed this in my first playthrough, but what does it mean when an IP box on the map has a box around it? Or the dotted box like around Uplink IPs.

A dotted box indicates that you have an account on that machine that does not grant administrator privileges, and a solid box indicates that you have valid admin access. Having an account on a machine that you use in your call bounce buys you more time on the active trace, more so if the account gives admin access. The only problem is that while you can get admin access fairly easily by getting in and out very quickly and leaving as few traces as possible, it will almost always gets revoked the next day; that can get a bit infuriating in cases where you covered your tracks perfectly, but it's not a major problem as you can use other tricks to make your call bounces more effective.

Is there any reason not to bounce a call off as many servers as possible, even if you don't have access (or it's a public access server)? I'm guessing it has something to do with bandwidth, but would like to know for sure.

You can bounce off of as many as you like...the more the better! :)

In fact until you get a monitor bypass, I would suggest going into the InterNIC and looking at the browse/search section. Then just go through the list adding as many as you feel like to your map.

Then just trace out the bounce path and save it. From then on, you can load the bounce path then just click the last link to whichever server you're trying to access.

Public access servers are useless to bounce through because they have no security. It's really not worth the effort of even adding them to your map, as a trace will go through them almost instantly. Internal Service Machines, on the other hand, are great even if you have no access at all, because the more security systems they have (proxies & firewalls) the longer it will take. It's even better to route through Mainframes and LANs, though usually I don't bother; I just stick with ISMs. Like A_Future_Pilot suggested, just add *everything*. Once you get lots of bounces through lots of machines with user and admin access, you can get 400+ trace times even when you're hacking mainframes.

Call bouncing tricks:
1. Register an account at all the banks and at the Stock Market. It really helps even if you never use them. You should do this as soon as you finish the test mission.
2. In most of my games there has been at least one ISM that never reset its password, so once you get admin access there you'll keep it forever. (You'll have to find it yourself, though, as it changes every game.)
3. Prioritize copy database missions, as they give you access to fileservers. The fileservers are unhackable because they don't have an admin account, so that's the only way to get access to them.
4. If a "trace a hacker" mission starts at an ISM, you'll get permanent non-admin access to that particular machine.
5. The Uplink ISM never changes its admin password, either. It's a pretty tricky hack to make, but if you manage it correctly then you'll not only get an extra permanent admin account, but you'll also get access to an extra fileserver later.
6. If you have a particularly hard hack to make and you don't have great equipment yet (say you need to crack an elliptical cypher but don't have a lot of processor power, or you need to break proxy+firewall but can't afford the Connection Analyser) just crack a few ISMs immediately beforehand. Your clearance will get revoked in 8 hours or so, but you should have more than enough time to get your job done. This tends to clog up the mission screen with a bunch of extra "trace a hacker" missions, but that's hardly a problem.