You will have to go to the dealer to put your dog crate in the trunk of these SUVs because there is not really a way we know if it fits in these cars. Trunks are shaped differently in different cars even though we know the cargo volume. Pretty much any car will hold your dog crate; it is just how many seats you need while carrying your crate. Although minivans will suit you so much better (because of large cargo space, lower liftover height than SUVs, and general practicality), I'll give you SUVs with a low lift over. The Honda Odyssey and Toyota Sienna are the best cars for you.
My first SUV suggestion is the 2013 Toyota RAV4. Legendary reliability, could definitely fit a dog crate, long list of standard features, and decently comfortable.
The Kia Soul has a low lift over being a wagon. Very practical, good on gas, 10 year/100k mile warranty (under certain conditions CPO, dealer, etc.), not as comfortable as the RAV4, reliable, good looking.
The Mazda CX-5 is a fun choice. Not as practical as the other two, but is reliable, fun to drive, good looking.