Well, in order
1: You don't really need a special backup program. Any CD/DVD burning program will work just fine. All you need to do is burn a copy of important files to disc. External hard drives are another option. The main thing though, is you don't really need some special backup program.
2: No, but one thing to do to see if it's a hardware or software thing, is to enter into the BIOS setup program. At bootup, somewhere along the initial screen will be a message about pressing some key to enter setup. Usually it's F2 or Del, but other systems may well be different. After entering into this special area, just let the computer sit. If after a minute or so it hasn't rebooted, it's a software problem almost for sure. If it still reboots, then you have a hardware problem... Most likely an overheating CPU.
3: If not IE, then your choice of Firefox, Opera, or Safari. I know New Zealand doesn't have the same all-you-can-eat style Internet we do on this side of the Pacific, so you will probably want to do a little research into each before deciding on which to download. The basic differences are: Firefox is probably the slowest of the bunch, but it has a pretty good selection of extensions. Similar to ActiveX controls, only a bit safer, these add new features to the browser. Some improve security, others help deal with common annoyances on websites, others add new functionality. Opera is probably the fastest browser out there, and it's an interesting little browser, but it lacks some of the highly useful features that can be added to Firefox. It's a good solid middle of the road choice though. Safari is the newest browser of the bunch. Apple created it a couple of years ago when it became pretty clear Microsoft wasn't interested in really developing IE for the Mac. Then, just recently, they decided to make a version for Windows users as well. Safari is based on a very lightweight rendering engine, which can be quite fast and nimble, but doesn't always render pages correctly. It has gotten significantly better since it's initial release, but it's not quite as good as Opera or Firefox at dealing with poorly constructed websites. But do some research of your own to weigh the various pros and cons of each, and decide for yourself. All three are free, and even downloading all three shouldn't put a significant dent in your download cap.
And to answer your other question... For the most part, yes, all websites are available to all browsers. There are a few that will ONLY work with Internet Explorer, and I usually promptly leave those sites in favor of one that will work with Firefox. Every now and then I have no choice but to use IE, but I always make it a matter of last resort. I will often complain to the person responsible for the website, though a lot of times I'm not telling them anything they haven't thought themselves. They are just bound by stupid corporate policies. Which isn't to say there aren't some people out there who are just too lazy and/or incompetent to know any better.