Yes you can continue to study but you will need to keep the school international student officer informed about your process. i also recommend getting help from an attorney because of the issues unique to LGBT couple. since you are in Alabama, your interview will be held in Atlanta. i agree with the others regarding the travel document.
I agree that the NVC will be looking for your original birth certificates and marriage certificate from your country, police clearance from the UK and any other country you lived from age 16 (unless not available). probably worth getting some legal help if you have complications such as you've mentioned. and with lottery visa processing, having things timely submitted is important so you don't miss the window of opportunity.
USCIS has been moving cases to local offices to speed up the processing for spouse cases, so hopefully you'll see faster results. however some cases get transferred if they want to interview you, that's possible as well.