There are those here with great knowledge about ML shotguns. I've enjoyed them about 60 years for hunting, not competition - originals and modern-made. Once you've determined the old girl is safe, I suggest a couple of steps/cautions learned "the hard way".
1. Before even sticking a screwdriver in a slot, be sure the tip fits the slot perfectly. Old metals can be quite soft, or brittle. I use a bit of "Kroil" to lessen the chance of buggering a slot or worse, breaking one off. Some screw slots require a chisel point, others a flat point. Replacing damaged screws on antique guns is hard.
2. When I decide to shoot an antique, nipples are of importance. Some have been open to the point they allowed far too much gas to come back towards the face of the hammer - others clogged so bad ignition was spotty at best. I replace (but always save and label) old nipples with properly fitting modern ones. If the nipples have been replaced by an owner before you or, as you remove them find all the threads do not engage, WATCH OUT.
3. Don't be surprised to find the right side lock and barrel more heavily worn/used than the left. This is really common among the old double guns I've owned. Looking back on how I shoot when hunting, it's mostly the right barrel that gets shot the most.
4. You have a really nice antique shotgun judging by the photos. Enjoy it. Thanks for posting.