Exploit:JS/Blacole.CE and Exploit:Java/Blacole.DW

Yesterday MSE detected this type of exploit and this morning it had detected two of them now. All have been removed and my computer is safe, but since there are no entries in the Microsoft online database, I want to know what excactly do these viruses do?

I'm not really worried as long as they aren't keyloggers or trojans. They're ranked "Severe" by MSE and I think I just have Java installed itself. I don't know excactly what Java version I have, but I don't want to have to remove it because I find it rather important.

Looks like a bad case scenario
Remove Exploit:Java/Blacole.DW (Removal Guide), How To Remove Exploit:Java/Blacole.DW | Cleanpcguide.com
Our security experts often suggest a clean reinstall with rootkits.

They've already been removed, it's fine.

But I still need to completely reinstall Java. Is it even safe?

Java has always been a source of potential problems.
The security people have said that rootkits are rather dangerous and you can never be sure that they are completely removed. I am not suggesting that your reinstall, I merely present the facts as I know them and leave the final decision to you.