Can you have 2 completely independent instances of Eclipse running at the same time?

If you want two instances accessing two workspaces, just run a second copy of eclipse

If you want two windows accessing the same workspace, you can create a second window with Window -> New Window

Yes, you can. Otherwise, no one could develop an Eclipse plug-in.

You have to uncheck the default workspace on startup. You can't have two instances of Eclipse referencing the same workspace.

Yes, you can have two instances of Eclipse running at the same time. However, they can't both access the same workspace at the same time. If you try to do this, the second instance will complain about it on startup and ask you to choose a different workspace.

Also bear in mind that Eclipse tends to be a very memory-hungry application. Running two at once may put a significant strain on your system.