Congratulations on the acquisition!
Your back sight is an original Lyman, and would be correct for all but the latest WWII M1928A1s. Some early WH guns had these factory installed. Bluing is the correct original finish, with the "ladder" (the part with the range markings) left bright.
Most WH guns require at least some tuning to function properly, so it's hard to know whether you have a gun or a drum problem. WH guns are nortorious for improper dimensioning in the feed ramp area, and some of the machineing operations on Colt and GI guns were omitted elsewhere for economy of production. Fortunately, the steels used at WH were good, so your gun will stand up to normal use just fine. Some guns run right out of the box and others won't fire more than a couple of rounds. All WH guns, though, will benefit from some expert attention. Many owners send their guns off for this work before they have owned them for very long. Has any work other than parts switching been done to the gun that you know of?
I would not take your gun to anyone without extensive Thompson experience, as they could easily do more harm than good.
Make sure your drum is properly wound to 9 clicks, or if your gun seems to have a very high cyclic rate, try the full 11 clicks.
There's a good chance as well that the rotor in your drum hasn't been properly greased since 1940, and there are members of the board who can provide this service. Using a special fitting, grease is forced into the proper places. It's not a job that you can easily do yourself. Once done, the drum is good to go for years!