I am pursuing a master part time in my home country. My job is sending me to Houston for 2020, so I will have to pause my studies. I wouldn't like to stop studying next year so I wanted to know if I could take one or two courses at a Houston university given I comply with the academic requirements. Hopefully, I can get credits back home but that's not a requisite. Still, I would like to know if I would receive an official proof for having passes the courses.
This is possible at many universities. You will have to check with them about enrolling as a non-matriculated student. There will almost always be a charge for tuition, but you can enroll (many places) as a student and get a grade. It is also possible in many places to enroll for non-credit, but you probably want the "credit" option.
Whether a university back home will accept them or not is up to them and not guaranteed, but, again, many universities are willing to do this for a limited number of credits. It might be best to check there as well so you avoid surprises.
As Buffy said, most likely, yes. Back in my days as a teacher, it was fairly common for me to do this (although the courses were usually tied to some sort of program or continuing education). Mostly, the universities are happy to get your (or your employer's) money.
As for official proof, you should be able to request an official transcript, even as a non-matriculated student. It may or may not come with a processing fee (on top of whatever tuition you would pay just to take the course).