maipenrai, The older rigid mountain bikes make excellent touring/all around bikes, IMHO.

What to check? Depending on your bicycle mechanical skills either gut the bike for a complete inspection and re lube or have the bike shop do this.
Drop handle bars? Least expensive option: Buy some that you like and use bar end shifters. Ride around to determine what you'll need to get the fit right before you install bar tape. You may need to change stems and an adjustable stem may help. Because drop bars have a different diameter than flat bars you'll also need a set of brake levers, cables and cable housing.

I used tires similar to what you describe on my first mini tour using my mountain bike and they worked better than I expected. There are also racks that can be used on a fork that lacks a mid mount.

Have fun.

Brad