Honda CRV, don't need to do any drivetrain disconnects to tow four down, whether 4WD or FWD, manual or automatic. Problem with those is your target price, CRVs don't start getting that low until you get back to late '90s models with more than 200,000 miles. Sellers are still asking more than $5000 for second generation CRVs over 200,000 miles, $8000 to $10,000 at 150,000 miles or more.
Chevy Tracker, you want 4x4, which most of them are. They are under $5000 for 2004 and earlier, 100,000 to 150,000 miles. They are harder to find than CRVs, has to do with how many sold, how long they last.
Jeep Cherokee, you would be looking for 4x4, just put the TC in neutral. I'm finding '95 to '01 at 100,000 to 150,000 miles just under $5000. 2WD sports are a lot lower, because most buyers in this market are looking for 4x4.
Jeep Liberty, replaced the Cherokee in '02 model year, 4x4 '02 and '03 have dropped under $10,000 but I think you aren't going to find one at $5000 yet unless it has some real problems, needing more money put into it.
Grand Cherokees will work, depending on which 4x4 system, there was more than one option for a number of years. Your $5000 price suggests 2001 and earlier, 150,000 to 250,000 miles. Low mileage Grand Cherokees are getting a lot more money, even model years before 2000.
That's looking to fit your price point, and your qualification of 4x4 and a cargo area behind your seats for the dog.
Best all-round for me is a different category. I want the kind of car I prefer to drive, but towable: compact or subcompact hatch with five seats, back convertible to carry more cargo, 30 mpg or better most of the time. I'm OK with, actually prefer, manual transmission. With automatic, this class includes Chevy HHR, Honda Fit. In manual transmission only, include Pontiac Vibe, Toyota Matrix or Yaris, Nissan Versa, Hyundai Accent, Kia Rio. The PT Cruiser also works if you can find it with manual transmission.
