The word 'Texans' is a noun, meaning 'Texan people'. The singular form 'Texan' may be a noun ('She is a Texan') or an adjective ('She is Texan').
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Both. Texans is plural. Texan's is singular possessive. If you're talking about more than one Texan, it's Texans. If you're talking about something belonging to only one Texan, it's Texan's.
Yes, Texan should be capitalized.
The South Texan was created in 1925.
The Daily Texan was created in 1900.
The desire for Texan Independence from Mexico.
The duration of The Lucky Texan is 3300.0 seconds.
The duration of Return of the Texan is 1.47 hours.
The duration of The First Texan is 1.37 hours.
Santa Anna was the military leader captured by the Texan army.
A Texan-Mexican or Tex-Mex for short.