personally I like the idea of the over-sized straight load because we had 1 pony and were looking for a horse for me so didn't want to find the perfect horse to then be limited by the trailer (one of the trailers was a slant that had very short stalls in it, bit older)
BUT given that you're talking rodeo, i'm guessing avg size quarter horse build so I would say a 3 horse slant, use that extra stall for more hay on long trips, return the favor that others have lent to you of picking up their horse while going camping ect.
and then something with as much or little LQ as you and your husband want, a good awning with some side-walls/tarps could greatly increase your usable space too.
As far as truck, what brand has the best service in your area... I hate ford because we had a lemon and terrible dealer when I was young (We= my parents), My dodge was a piece of junk (assembled in Mexico, best friend's assembled in US and it survived a lot of abuse), current Chevy is great (local chevy dealer a pain, but I take it to the GMC dealer and they are great) If you are towing a lot I would certainly go with a diesel, 3/4 or 1 ton depending on trailer probably want a long bed so you can get a nice wide nose gooseneck for a bed in the LQ, Manual transmission unless you go with the Duramax Diesel then you are stuck with an automatic but the Allison has been awsome for my daily towing as well has hauling the horses.