Some U.S. models did, I believe the Germans didn't but they had exits on the side hull on some early models PzIIIs. The Soviets had them on some models. As far as I know it wasn't really a common practice.
I saw "The Beast" - great movie.
"Do all tanks have an exit underneath the body?"
I'm not an expert, but I think that at least most Russian tanks have it.
A view into the forward fighting compartment through the driver's (right side) escape hatch. Visible slightly right of centre is the driver's seat back and the flame gun assembly further beyond that. There is a surprising amount of room up forward, in contrast to the constricted turret space. Churchill tank.
Here is the belly escape hatch that was added to the later M5A1 hull directly behind the assistant driver's seat. As I mentioned earlier, the first Stuarts had the reputation for being death traps for the two drivers. It was next to impossible for them to get out through the early turret basket design if either driver was injured and their over-head hatches were fouled by the turret and gun. So this floor escape hatch was added as a safety design change for a quick egress in case of emergencies and it was used. The large rectangular hatch is shown here with the assistant driver's seat back removed for improved clarity. The characteristic wide tan or green seat belt webbing with steel buckles is seen, attached at the rear to the seat back bracket, but seat belts were generally removed before going into battle. By lifting the escape hatch's long release handle, the hatch would fall down and away, hinged at the rear by two pivots. The tubing seen at the far left in the photo is for adjusting the height of the seat for driving head out of the over-head hatch, the paired long springs assisting to raise the seat when adjusted. Even though the over-head hatch-mounted M6 periscopes could be rotated and pivoted up and down, the driver's view when "closed down" was very restricted, and head out driving was his preferred style whenever possible. M5A1 Stuart light tank.
Escape hatch under seat. M103 Heavy Tank
Sherman, Matilda, M24 Chaffee tanks had escape hatches in the floor, usually under or behind the driver's seat. Tigers and some others had them in the side of the hull.
There was an escape hatch under the driver seat of the M60A1 tank. Most of us tied the handle closed, since we weren't expecting to have to evacuate the tank in this manner, and the hatches had a habit of falling out. They weighed a couple of hundred pounds and were a b*tch to get back in place. Fact 2: In the rainy winter weather of Kentucky, if you drive ten tanks over a piece of ground every day for a few weeks, you can create a mirey muck with the consistency of modelling clay. Stack them together: Tank drops escape hatch. Young instructor has trainee crew pick it up and throw it in the storage rack on the rear of the turret with the intent of reinstalling it at lunch. Driver training continues. Tank comes off a little knoll into a ditch full of the afore-mentioned mud. The nose pitches downward. The mud moves, most of it outward, but A BUNCH squirts up through the escape hatch port, into the driver compartment, and surrounds the poor driver. They had to dig him out. He was unhurt, but cold, wet, muddy, and several shades of blue. It took shovels to get the poor kid out of there.
Those driver's escape hatches were a huge pain in the butt. We found it worthwhile to carry a hydraulic jack so nobody had to bench press it back up.