Should have MADE me buy this game when I first signed up! I love IT!!!! Just a few questions....
First of all it would help if you told which game you are talking about. From your questions I presume that you are talking about Duke 3D.
1) Is there any way to minimize this fucking DOS lag?
I personally don't use the GoG version of Duke (yes, I'm a horrible person.) but an original copy (darn it was hard to get my hands on...) with eDuke so I don't really know how Duke behaves on dosbox. I think that you can use the files in the GoG version to install eDuke or other similar version of Duke that is compatible with modern OS. I can't help you with the Dosbox though. Sorry.
2) How do I install the mods?
Most mods come with installation instructions and I think you should be able to follow them even with the GoG version. For the HRP, just get the duke3d.grp and follow the instructions.
I personally prefer the original Duke 3D instead of HRP though.
3) Uh........Is there anyone playing online?
Sometimes, if you know where to look...
4) Why is it that when I try to launch the game with the shortcut it fails but when I do via graphics settings it works fine?
Sorry, but I don't quite understand what you mean.