We also are moving from CA to TX. In the past 5 years I've made the trip roughly 25 times. There are no stops where your vehicle is searched. There is one that looks for illegals but all they do is ask if your american. If you look hispanic or have a foreign accent then you will get probed in more depth. There is nothing relating to animals of any kind on your trips, except for commerical truckers.
If by chance your taking the 10 freeway: exit 200 in arizona has the best gas/diesel prices, then exit 37 in texas (el paso). For lodging, exit 20 or 22 in Lordsburg, NM or Demming, NM (don't remember exit #). It's about $35 for a single individual. Not the classiest places, but comfortable enough for a sleep over. The way I travel (truck and 16' trailer) I make it to Lordsburg or Demming the first day depending on when I leave in the morning. I get to my place just outside austin on the second day (~ 1,400 miles).
I would suggest taking the Phoenix bypass (exit 112). It is a side route which, in total, is about 8 miles longer in distance, but about 30 minutes shorter in time depending on traffic. If you chose to go through Phoenix try to time it where you miss the rush hour traffic (10 to 2). Same for El Paso.
Rest stops are your friend. Try not to make your trip during the summer months.
If you need any more information I'll be happy to provide it.
Good Luck.
Ron