Do they cut this gag out of TV airings?
Probably...but then, I haven't seen this one on tv in 15 years.
Elmer Fudd getting strangled in midair by a "Alcatraz Ascot" in Design for Leaving (1954) is still intact on Cartoon Network/Boomerang, so I don't see why they would have a problem with this particular gag.
Do they even air Tex's cartoons on TV any more?
Sometimes. Cartoon Network, for what ever reason, aired "The Counterfeit Cat" and "Ventriloquist Cat" during their Tom and Jerry run every weekday (also the Filmation shorts).