To my knowledge the only way to get STD is to go out on FMLA. Unless you want to take vacation/sick days. For FMLA and STD there's forms your dr needs to fill out and give a date when you can return to work. You get 12 weeks of FMLA a year (Not calendar year, from the time you first take it) and after that they can fire you.

That's how I got fired from my first job. The rules might've changed though, I haven't worked since 2015 so not up with all the working laws. Yes BP is covered under the ADA and is an illness you can take time off work for.