I installed Apache2 according to the instructions in SuSE 10.1 Help files which state:
1) Start YAST
2) Select System + System Services (runlevel).
3) Search for Apache2 and enable.
Then it says that the web server starts immediately and that by saving your changes with 'Finish', the system is configured to automatically start Apache in runlevels 3 and 5 during boot. Now that is my understanding of what looks a rather simple process because it is now more highly automated than before.

Help files say that Apache can be started (shell) by 'rcapache2 start'. Does not work for me. Nor does the traditional variations on 'start' work. When I use CHKCONFIG (without options) I see that Apache2 is there, but OFF. I also checked that it indeed is operating under runlevels 3 and 5. I suspect that the problem may be with the Firewall as port 80 should be open, and I cannot see how to check or alter that.

Lastly. Why do I need Apache when all I want to do is run MySQL + PHP?
Can MySQL not run on its own?
And does an Apache installation open a back door in a system?

The firewall is not the problem. If it "locks" port 80 apache would start fine but you can't connect to it. You need Apache (or any other web server) because to see the PHP pages there must be of course a webserver.
Maybe you could investigate the logs looking for problems (did you run rcapache2 start as root?).
Apache could open a back door in the case the port 80 is open from anywhere, so anyone can connect to it and open browse your php pages/ecc if they aren't configured properly.