I recently bought a BM-800 condenser microphone set and as i'm hooking it up, I want to be able to use my headphones with it, but the split USB input and output won't allow the mic to work in the first place, let alone attach some headphones. I've tried messing with all sorts of settings but i'm just not as skilled with computers as other people, PLS HALP!
While the BM-800 is a "magical beast" that will already work with the asymmetric 3.5mm plugin-power input of a laptop (even though it can also function as a low-quality microphone on +48V phantom power on balanced XLR microphone inputs), it is not a USB microphone and there is no "split USB input and output" on a laptop: please refrain from using technical terms in your description when you don't know their meaning.
Trying to replace your description with something that makes more sense is of course mere guesswork but it sounds like you have a combined headphone output and microphone input on 3.5mm: those are common for headsets and will accommodate a regular headphone without microphone equally well (shorting out the microphone input in the process).
As to the screeching sound when plugging in just the headphones: in that case your computer likely uses its builtin microphone and your current settings make it hearable on the headphones where the microphone will pick it up again, a vicious circle known as feedback. Check for controls switch off your headphones or just put them on (then they should not transfer enough noise back to the microphone to matter).
For using a microphone separately, there are adapters splitting the combined 3.5mm socket up into two separate ones. A look on Ebay shows one at minimal price (though with delivery from China, not exactly fast) with the keyword bomb "3.5mm Stereo Headphone Microphone Audio Y Splitter Cable Adapter Plug Jack". I'd expect some similar keywords to yield similar results, and these adapters are standard fare and should also be available at pretty much any brick and mortar store selling laptop accessories.