So upon hitting the "play" button in the launcher, before the game loads but after the launcher closes, suddenly the launcher pops up again, without a "Minecraft has encountered an error and needs to close" message.

By changing the settings to keep the launcher open, the game simply never launches. When I press "play", it looks like it's loading and then just stops.

It did give a log in the development console tab, though:

http://pastebin.com/upmxp5LU

I have not installed any mods, just the Forge - Recommended file for 1.7.10. I already searched the forums for "return to launcher" and found nothing.

Are you sure there is no mods?

It looks like to me you installed The Aether mod. What is for 1.7.3 and not 1.7.10!

But you also need PlayerAPI for that mod.

Ah. That appears to be the problem.

 

I had just deleted the old Forge profile, and thought that would also get rid of the Aether mod. So this means that somewhere an old 1.6 version of The Aether is still within my game files somewhere, and I need to remove it before Forge can work. This would also mean that, since I re-installed the same version of Forge thinking that would make the problem go away, but also really just deleted the old Forge profile, that there are copies of the same Forge version all over.

 

Do I have this correct?

Profiles have nothing to do with files on disk, you have to delete the files on disk to 'uninstall' a mod.

So yes, you still have mods in your 'mods' folder.