A couple of weeks ago I tried to install java on my server, just to play with. It showed to be a mistake, since suddently I got an error on my active webforum. The erorror was a http 500 internal server error

I checked my error-log file for the page (there was no error on other pages on same server) and it said something with safe mode, so I disabled safe mode in php.ini. I thought it already was disabled but apparently not.

Now my main question is: what is safe mode for?

And bonus questions: if I was to turn it back on, then how can I find what is wrong with my web page when safe mode is turned off?

(bonus info: my web page is now reeeeeally slow for some users - but only for some. It's not browser dependend. And I don't know why).

I also seem to get this error in my error-log:

Can this have anything to do with my internal server error when safe mode is on?

It is better when safe mode is off. It is also deprecated on new php versions. It not give good security, but unnecessary limit what php programmers can do. Better is using good configuration and buitin Linux security technics, which anyway must be implemented on good public servers. For example users/programmers can get problems when uploading files by ftp and then trying to access it by php script - and common solution was chmod files and directories to 777, which is even less secure. Here you have full list of functions affected by safe mode.

These notices about already defined constants should not impact performance or safe mode. Safe mode affect "connection" between php scripts and filesystem.

You should post exactly what error you get when safe mode was enabled to get answer on your bonus question. Slow web pages for only some users indicate to baddly written scripts or sql server low performance (by bad queries or just to low resources). Too low details to conclude something more. You should check logs of sql server, mysql has something like slow queries log (it need to be enabled) and some other.