You can contact Thompson Center Customer Service through their website for help. They are still in business, you know. BTW, really good rifles.
You can download manuals from the Thompson Center website listed here in the related links, you can also find a phone number there to request a hard copy. You will need to know the model. In this case the 25th anniversary Hawken, same specs apply as the standard Hawkens rifle.
Most black powder bores have a twist that is relatively slow. Say 1:36 or 1:50. They were 1:48 twist in 45 and 50 caliber
Thompson Center has no official published serial number list, but I can usually tell a general age by the model and serial number. The model is in most cases engraved on the barrel, ie Cherokee, Seneca, Hawken, New Englander, Renegade etc. and the serial number is stamped on the barrel itself or on the breech plug. You can also contact Thompson Center directly and they may be able to help.
Sure. I like pictures of Hawken rifles.
50-350 depending on specifics
If it is a Thompson Center Cougar it is meant to fire a patched round ball or mini ball style round, (conical lead bullet aka maxi hunter, buffalo bullet etc.) With the relatively slow twist of the rifling 1:48 it will fire Sabot bullets but will not be as accurate at ranges past 125 yards, Maximum powder charge should not exceed 100 grains. All in all it is the same mechanically as the Hawken .54 and I believe you can still download the manual at the Thompson Center website.
Read books, look at real ones in museums, gun shows, gun shops and ask Hawken collectors.
You need a professional appraisal. Fakes abound and are vey hard to spot if you are not familiar with the item in question.
100-300 USD
Cabellas still sells one of the versions Sile sold, If it has the round double ring patch box. If it was an older one with a very close look to the Thompson/Center rifles it was made by ultra hi, a Japanese company. Investarms made rifles for sile in the 70s and 80s they were imported from Italy.
You will have to call TC to find out.
If its a real Hawken, and most are not, then its almost priceless. The vast majority of the real authentic Hawkens are in museums or private owners who can trace their history back 150+ years.