So, obviously tanks were used extensively in the Second World War. Tanks, of course, travel on tracks. What happened if, say, a tank's track broke in the middle of a battle, due to a shell hit or landmine or something.
It seems that both Soviet and German regulations were fairly specific as to when a crew were permitted to un-ass the tank. In the German case, the tank had to be on fire. If immobilised but not on fire, they were to continue to fight.
The American manual gave a little more flexibility to the crew. I'll come to this in a second.
The first problem a crew needs to figure out is "what hit us?". You're driving along, there's an almighty "Clang" and then you either hear the track sliding uncontrollably off the rollers, maybe, or perhaps the tank slews to one side until the brakes can be applied. Was it a mine? Was it an armor-piercing shell which hit the track? If the latter, is a second shot going to follow on in about three to eight seconds, and what might it do then?
If the TC's hatch is open, he may have a clue, possibly a large cloud of dirt or some such. Assuming they know it's a mine, the next question is to estimate damage. They are now in a minefield, and stuck, which is not a good place to be. Minefields, if they are laid by someone competent, are going to be under observation, and probably in short order, either direct or indirect fire.
Then they must evaluate the extent of the damage. The first thing is that the driver's going to gingerly try driving to see how the controls respond. Let's say it's confirmed, the track's not going anywhere. So a look must be taken. You don't want to just jump down off the tank, though. If there's one mine, there may be more. And those 'more' may be anti-personnel mines. It would ruin your day to jump down off the tank and onto another mine.
If it's the right side track, the TC may decide to take a risk and lean over the side to take a gander. If it's the left side, it might be difficult depending on if there's a loader's hatch in your M4.
Let's say you've concluded that it's a mine. If it's not the middle of the battle, you probably want to just repair it. Best way out is to have another tank nose up to your rear, following your tracks. Hook up the tow cables, if you have to get on the ground to do it, land on the path the tracks took, and then reverse the tank back out of the minefield and behind cover.
If it is the middle of the battle, or your track was blown off by an anti-tank shot which missed the hull, you have to make an extremely rapid decision. FM 17-67 states suitable occasions for abandoning the tank are if the tank is on fire, or if it is disabled and has become a vulnerable target. It is not an exclusionary statement, it just says "ordinarily". It is presumed that the tank commander has a brain and can make a judgement call of his own. A stationary target is definitely a vulnerable target. Much easier to hit.
Depending on his judgement of the situation, and it may have to be a snap decision as soon as the shock wears off, he may have the crew stay in the tank and fight it, or abandon the tank. He has to balance, in about one or two seconds, the choice between what his tank can still contribute to the fight, the chances that a follow-on shot from a gun which obviously has the range will do more than just knock off a track, or the risk of small arms or fragmentation injury to the crew in the event of bailing out.
The one thing they are highly unlikely to do is to repair it in the middle of combat. Repairing a broken track, in ideal circumstances, is going to be a 20-30 minute job. A battlefield is not going to be an idea circumstance, it'll probably be muddy, the tracks may have snapped in an inconvenient location, and, oh, people are shooting at you. /u/TankArchives example of Maria Oktyabrskaya is a case in point. She may be a Hero of the Soviet Union for chosing to repair her tank in a battle, but she became a dead Hero of the Soviet Union. Dead heroes were not considered to do the US Army very much good, and it would take a fairly incontrovertible circumstance (or repeat offenses in questionable circumstances) to brand a tank commander at fault for bailing out of an immobilised tank. If the choice is losing the tank's firepower due to abandonment, or losing the tank's firepower because the crew stayed in and the tank was hit again until it burned (common, given that tanks which are not burning are not obviously out of the fight) and losing the firepower and the crew, the decision is fairly easy.
I cannot speak to the British policy.
I can speak on how battlefield repairs were handled in the Red Army.
There is one notable exception to this. The A-20, a precursor of the T-34, was designed with the same gear ratios for tracked and wheeled drive. The tank could drive successfully with just one track. However, this tank was not mass produced, so only one tank in history would be able to do this. The A-20's successors did not have convertible drive, its precursors had a different gearing so it would not be possible to run one side on wheels and one side on tracks.
Yes, that is the only way to do it. This was considered an exceptional act of heroism, as it was very dangerously. One of the most famous tankers of the war, Mariya Oktyabrskaya, repaired her tank under enemy fire but was fatally wounded. For this she was rewarded with the highest honour offered in the USSR, the title of Hero of the Soviet Union. This is far from the only case. Guards Senior Lieutenant-Technician L.L. Sukstanskiy was awarded the "For Courage" medal for repairing his tank under fire, Guards Sergeant V.F. Gryazev got the same medal for remaining in his tank when it was disabled, spending 10 hours inside under fire, and then repairing it and driving away at night. This leads me to question 3:
No. Abandoning a tank that could be repaired was frowned upon, as if the enemy got their hands on it, they would very likely either tow it away themselves or deal so much damage to it, that it would not be repaired. Leaving behind a functional tank was equated to sabotage and punished harshly. Again, it was acknowledged that this is no easy feat, and crews who did this would be rewarded. For instance, the crew of Lieutenant Aksenov defended a knocked out tank for several hours during a battle to close the Korsun pocket, as a result of which they were rewarded with the Order of the Patriotic War 2nd class, even though they eventually retreated and the tank was lost.
P. Samsonov Designing the T-34
https://pamyat-naroda.ru/heroes/podvig-chelovek_nagrazhdenie150024079/
http://www.podvignaroda.mil.ru/?#id=18284723&tab=navDetailManAward
http://www.podvignaroda.mil.ru/?#id=19195104&tab=navDetailManAward