Payment to underage offspring

Sure. Whenever they're old enough to do work worth paying for, and as much as you'd pay an unrelated person to do the same (quantity and quality of) work. My 6yo is incapable (primarily due to unwillingness) of doing office work, but if I needed print advertising with kids, could "model" for that. Paying your 4yo $5,000 a year to empty trash cans is not going to fly. Paying your 16yo $8 an hour to answer phones / make copies / file documents should be fine.

In general, the Wage & Hour people and the Work Comp people will not get involved in family disputes. If your situation is such that you would be concerned about either wage & hour or work comp exposure with respect to your minor children, whichever agency oversees child welfare IMHO ought to be of more concern to you.

If the kids are performing services for an unincorporated business owned solely by their parent(s), there's no FICA or FUTA. Depending on your state law, you may or may not have SUTA exposure, but it's unlikely you'd have any unless you had other employees. The kids still get W-2s, not 1099s, because they are your employees, not independent contractors.

Disclaimer that I'm not an attorney, nor do I have any familiarity whatsoever with Texas laws. If you want something you can depend on, pay a Texas attorney for their opinion! :)