Can you still run the bot while using the no cd method with d2 loader?

Yep

However, it is recommended that you don't use any third-party programs along with mmbot.

D2 loader is referenced in the mm.bot so I don't think it is being counted as a 3rd party program. I use it and have never had a problem; in fact I think you have to have it to run multiple keys (but I only have one so I might be wrong).

No it doesn't require that for multikeys., but users tend to experience Diablo II internal errors when running it with mmbot.