1. No licenses needed for rifles in TX. You don't need any licenses to own long guns (rifles and shotguns) or handguns. The only reasons you'll need a license involving guns down there is if you plan on carrying concealed, you plan on hunting with it, or you plan on getting something fully automatic. (That's a Federal thing, very expensive, lots of background checks, and fully auto weapons are very expensive.) Oh and handgun wise you are no longer limited to something with a 4.2 inch (105 mm) or longer barrel like you were in Canada.
Like FG said, you do need to pass a background check, which is complicated by the fact you're not citizens (I'll check the requirements, but I think you have to be in the country for 6 months before you can buy one), but overall, no felonies, and no violent/domestic abuse misdemeanors and you're fine.
Oh FG, you can skip the background check if you have a CCW in GA? CO they don't allow that, it's every firearm sale, bar none.
FWIW as far as I know, the only states that require you to have a license for long guns are in the Northeast part of the country.
2. .303 is the old British round from WWI and WWII, and typically not a very popular round over here in the States, so yeah it's pretty expensive to find. If you think that's bad though, look up some .50 BMG haha ($3 per round is the cheapest I've ever seen it). The 7.62 is also a fairly large round (.308 equivalent). If you're looking for something that's cheap to shoot, you can't beat a .22. Plus it's accurate and if you decide to take up hunting, it's a good small game gun.
Have owned - 2001 Suzuki Volusia
Current bike - 2005 Kawasaki Z750S
MMI Graduation date January 9th, 2009. Factory Certifications in Suzuki and Yamaha