I was a reserve MST who also worked in the maritime industry as a civilian. I don't know what may have changed since I retired, or how things work at Mobile, but I'll answer some of your questions to the best of my knowledge.
"Do reservist MST conduct on-board inspections?" There are R-PAL billets for MST vessel examiners. Where I drilled, we had a team of Reservists who conducted the weekend's Port State Control exams and manned the Vessel Arrivals Desk processing notices of arrival and scheduling exams. For a Reservist who isn't prior active duty, getting qualified can be difficult and take a long time, but it is possible. It's difficult, because proficiency is gained through repetition, and there aren't really enough drill periods in a year to get the repetition needed. With General Mandated Training, various all-hands evolutions, and various administrative items taking up drill time, combined with whether or not their are any inspections due on your drill weekend, Reservists aren't likely to participate in inspections every drill period. One year, one of my people was only able to participate in six inspections. It's tough to get proficient without spending a lot of time in the field. The best chunk of experience comes when you are able to conduct your annual ADT working directly with/for the Active Duty inspectors. You'll get on more ships in two weeks than you will the rest of the year.
"What is drill like for a MST?" It depends on your billet. Reserve MSTs can have the opportunity to participate in vessel safety inspections, container inspections, facility safety inspections, pollution response and investigation, vessel and facility security inspections, and quite a few other marine safety activities. You might get sent to hand out brochures wuth the Auxiliary during Boating Safety Week. You'll spend some time in front of the computer doing General Mandated Training, but I hear that the amount of GMT has been decreased since I retired. Twice a year, you'll weigh in with admin. Once a year, you'll spend the morning signing forms to update your Record of Emergency Data and SGLI beneficiary forms. You will be frequently asked about your readiness (PHA, dental, vaccinations, etc.) Every now and then, you'll whiz in a cup. If you stick around long enough and advance, you'll have to watch SOMEONE ELSE whiz in a cup. Some weekends will be super-fun and fly by; others may be dreadfully boring and go by very slowly.
"Would working in oil patch interfere with the responsibilities of a MST?" Again, it depends on your billet. Safe to say that you won't be going out to inspect MODUs. Your chain of command will evaluate your billet and your civilian work for any conflict of interest, and may assign or reassign you accordingly. In my situation, I provided my Command with a constantly updated list of companies, vessels, and facilities that I have contact with in my civilian work. Everything stayed above the table, and I would recuse myself from any USCG evolution that would have contact with any entity I had a business relationship with. It was kind of a slippery slope, and I'm sure that arrangement wouldn't have flown in some Districts. But it worked out OK for me and my unit.
I had a blast as a Reserve MST, but I also know some folks who have found it boring, mundane, and uneventful.
I hope some of that helps. If you have any other questions, feel free to ask. Good luck!