I would say that - in general - you're right. "Entschuldigung" means sorry and excuse me. The only problem is that you use a fixed expression in german if somebody dies which is Mein Beileid or something like that.
Just two examples for the usage of "Entschuldigung".
1. to be sorry -> Entschuldigung
This is just for excusing if you broke something.
I'm sorry that I broke your window.
Entschuldigung, dass ich dein Fenster zerbrochen habe.
2. Excuse me -> Entschuldigung
You mainly use this expression when you want to ask a stranger for the way or something (Excuse me, where is...). You can also say Entschuldigen Sie instead of just Entschuldigung - that makes it sound more native...
Excuse me, where can I find the next restaurant?
Entschuldigung, wo kann ich das nächste Restaurant finden?
3. to be sorry for sth -> für jemanden leid tun
Now things get a little bit complicated ;). If you want to say that you're sorry for sth you are not responsible for you can use Es tut mir leid für dich.
I'm sorry for your bad grade.
Deine schlechte Note tut mir leid für dich.
I just have to mention that it sounds a bit strange in german, but it works and I don't see another way to express this.
I hope that I didn't make things more complicated for you ;).