Did I choose the right bottom bracket?
I'm only dimly aware of Thompson bottom brackets, but:

The visible thread on the spindle isn't a concern or a sign that something is wrong. My understanding is that, like Ashtabula/one-piece/American bottom bracket shells, the shell widths found on different Thompson shell bikes can have some variance due to either low tolerance or lack of real standardization, and the BBs have the threaded length to accomodate that.
It looks like the chainstays are in the 490-500mm range or perhaps longer. The front chainline being out by 5mm on a bike with chainstays this long is likely to be inconsequential. If there's no roughness or noise when pedaling, it will be fine.
This appears to be a coaster brake and/or internally geared hub. If so and if the cog has offset, make sure it's flipped out. If it's not offset and if it's the normal three-notch type, and if you wanted or needed to make the chainline better, getting an offset cog can buy you a few mm.