TexasCHLforum.com

Acually requirement is to be admitted to US on immigration Visa. See FAQ above. Now, l have feeling that DPS are providing conflicting info betweem packets req section and DPS website

Just a warning a federal judge can easily rule that hunting licenses or permits dot allow a person to carry a concealed while said person is not actually engaged in hunting or sporting activities

That is dodging the Federal rules, Federal rules may allow acquisition of firearms for hunting purposes (or sporting purposes) not for self-protection and certainly not for concealed carry purposes.
Straight from TX DPS website:
Eligibility Questions

Q)If I am a legal resident alien, can I get a license?
A)Under federal law, aliens who have been admitted to the U.S. under a non-immigrant visa, usually are not qualified to purchase a handgun and therefore, do not qualify for a license. However, as long as you were not admitted under a non-immigrant visa and are a legal resident of Texas, you may qualify.

Link: http://www.txdps.state.tx.us/administra ... hlfaqs.htm" onclick="window.open(this.href);return false;

It is very clear: Temp non-immigrant VISA, such H1B, L1, TN Status, F1 etc, are not eligible. Immigration VISAs, such wife immigration VISA, I-486 (change to permanent Resident), green card are eligible if there is no baring disqualification.
I would say, even if somehow a non-immigrant was issued a CHL, had a valid hunting licenses, and was caught carrying, Federal Authority (FBI) may arrest the non-immigrant and prosecute for unlawful weapon carrying charge.

In my packet, there was an extra white paper instructing me to include a copy of my green card.

Texas does not require US Citizenship to grant a CHL, they require a green card.

what shootthesheet said is correct. Since your friend is married to US citizen he can get his Green card and US citizenship on faster track than employment based immigration. He does not need to wait to file for a Green card.