Yes trailer inspection is required in TX other than little trailers under 4500lb gross. They'll look at brakes, lights, reflectors, tires, etc, put a sticker on, pay your $14.50, and you're on your way.
They'll have a sign like this out front if they do inspections.
Harris county is an emissions county, so you'll have an emissions inspection on your Suburban for $39.75, unless it is 25 years old.
There are some changes coming, no more inspection stickers starting in March. Starting then you have to get the inspection within the 90 days before renewing the registration, there will be just the one windshield sticker for the registration going forward. I haven't seen them yet clarify if this means the inspection sticker on the tongue of trailers also goes away or not. I hope not, it's easily spotted and makes it easier for officers to make a stop and get the junk with no working lights off the road that you constantly see roofers/contractor types towing around.