Roughly between 41 B.C. - 2012 A.D.
On 90% of the guns it is stamped on the right side of the barrel as "Thompson/Center Rochester New Hampshire" and "HAWKEN 45 Cal" along with a serial number on the left side. If not the easiest way would be to go to gunbroker.com or Google and search for Thompson center hawken, compare the pictures and description to what you have.
In a TCA, if it has letter preceding the serial number, it was made after 1971.
No way to tell from just the serial number.
It's a 54 caliber round ball
You will have to call TC to find out.
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.
Being a 5 digit serial number puts it early in production , most likely early 80's. However the Hawken was not produced by Thompson Center in .36 caliber, they did make a drop in replacement barrel in .36, but they are very rare. I suspect you may have the Seneca model, this you can determine by measuring across the barrel, flat side to flat side. The Seneca will measure 13/16" across the flats. A Hawken will measure 15/16". The Seneca also has a patch box that is rounded towards the but vs. the Hawken being pointed towards the but. If it is a Seneca .36 in 95% + condition it could be worth in the $500 to 700 range, the Hawken with the .36 drop in barrel would be in the neighborhood of 450 +/-
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.
TC guns don't have a model number printed on the gun per say, they have a model name (ie Hawken Renegade Triumph Tree Hawk etc.) There is a corresponding model number in the catalog 9503 is the Hawken caplock for instance, but if you need to identify your gun go to the online auctions like gunbroker or auction arms and compare your gun to the photos there.
Contact Thompson Center Customer Service at 1-866-730-1614
Thompson Center does not publish date ranges for there guns, but from the number I would estimate the date to be mid 90's
You will need to contact Thompson Center Customer Service thru their website.