I have not made any progress on Rudolph Bair. In answer to your question about Bair in Civil War, Revolutionary War, and War of 1812 I have the following information:
1) John Bair (1810-1884)wrote a letter in 1874 listing his ancestors. In this letter he lists that his father's brother rudolph was killed in the battle of Franch Mills, Canada.
2) I have a copy of Warrant No. 26660 where in John Bair (same as above) and his brother Henry file for bounty land under their claim that their father Adam Bair was a Private in the 19th Infantry in the War of 1812.
3) a copy of "Records of Men Enlisted in the US Army" lists Adam Bear as enlisted in Eroe Pa. and died at Williamsville Dec. 1814 attached to the 19th infantry.
4) My mother's research not only lists John's Father [Adam (1785-1814)]and uncle [Rudolph (1783-1813)] but also a Rudolph cousin of Adam as Bairs who served in the Army during the War of 1812.
5) John's Mother was Catherine Bowermaster. Catherine's brother Peter Bowermaster served in the army during the War of 1812 from 1 Nov 1812-25 Jan 1813 and 8 Aug- 8 Oct 1812 (BYU 929.377/1 OM3r page 45 Roster of Ohio Soliders in War of 1812). There were other Bowermaster's in the war.
I hope this is helpful. There are probably other Bair or Bowermasters in the War of 1812. Joh was in Utah before the Civil War so did not participate. I am having a hard time proving where the family was before 1783 (then in Westmoreland). They may have come after Revolutionary War. Them may have even be Hessians (but I doubt it). I have not been able to prove it, but I feel that the Rudolph who came in 1739 on the Lydia may be our Rudolph ( the father of Henry who was the father of Adam listed above). I hope this helps. Let me know if you have any other information that will help solve the mystery of the Bair's.