1. What are the main purpose and pros/cons of various tank hatch designs?
For example, many Soviet/Russian tank hatches open forward when Abrams' hatch opens backward to provide "Open protected position" while others have hatches that slide sideways etc, etc....

2. When a tank is hit and neutralized, what factors make its main gun to droop or not droop? I presume it's primarily because of hydraulics failure?

I assume you mean just the TC's hatches?

Position and opening its pretty arbitrary, but does have some logic to it. Soviet hatches open forward to provide cover to the front while egressing the turret. US and most Western tanks open to the rear to provide access to the TC's weapon.

Guns are pretty well balanced on their mountings but when damaged or while the vehicle is burning up, parts of the breech side can fall off, especally if the coax ammo bin is attached to it. This shifts center of gravity forward and as the hydralic or even electric elevation actuator is destroyed, the gun tips downward.

When I see pics of the Russian-style hatches, I notice how much it increases the silhouette of the tank, as well as forcing the TC to expose himself much more to be able to look around. It looks as if he has to climb halfway out of the turret to see past his hatch. With Western tanks, OTOH, he only has to expose the top of his head.

At least at T-72 TC coupola can be rotated so hatch is on the side or rear, in which case tc can just poke a head out. Actually, for firing a MG TC cupola has to be rotated backward, with hatch protecting TC from behind.

The Leo TC's hatch is designed so that it can be held above the TC's head in an open-closed position, permitting the TC to look outside without fully exposing his head. Because it slides open the mechanism can be manipulated from under armor, though it's somewhat slower than just swinging the hatch open over the top (like the loader's hatch).

Neither hatch has a vertical position like Russian tanks, so as to not obstruct the AAMG traverse or the PERI field-of-view.

Have you ever tried that personally?
It just doesn't work, neither with headgear on or even without it. Unless you have a head like Sid from Ice Age where the eyes are located near the top of an otherwise flat skull, the distance is just too small.

The design rationale of the Leo 2 commander's hatch was to avoid the silhouette expansion that the typical lift/flip designs involve. I would agree that the designers missed a great opportunity to create an open-protected mode. It would have been great to have it. But the way how they implemented it, it is unworkable.

The latest Leo 2s now have a massive sliding hatch for TC and gunner, about 160kg each. Even slower to operate, but safe against bomblet attacks.

Isn't that an Israeli innovation; the Urdan hatch? I remember reading that it (or something like it) was installed on the M48A5 and M1s--the early ones, at least.

Umberella hatch position was available to Conqueror cmdrs and also Centurion

It's not good for observation, I give you that (you get a better view from the periscopes). But it does allow you to slide it open just enough to use hand signals and peek outside without fully opening the hatch.

The gun droop is more to do with the balance as every tank gun I have ever worked on has been mechanicaly gears with a electric moter driving them the same is probably true of hydralics systems it's more accurate for elevation. The reason the gun drops or elevates is because guns are either breach heavy or barrel heavy. meaning that with a round in the chamber the ballance point is either over the barrel or over the breach. The reason gunners have to make their last elevation movement in elevation if it is barrel heavy or depression if the gun is breach heavy to correct for backlash in the elevation gears and the sight mountings(correcting for the wear in the elevation gears). If the tank suffers a Catistrophic hit and these gears are destroyed or the gun is knocked of their trunnions the gun will elevate or depress depending on the balance of the gun to breach.

There is no such thing as open-protected in Leo 2A4 as you can't lock the hatch open-protected or "umbrella" mode. It very dangerous too keep it on that position...when you drive to the first bump...
In fact it is strictly forbidden in peace time (only two positions: closed and locked down or open and locked down) in FDF and AFAIK also in Germany.

CV9030N/FIN TC hatch is very good to operate on "umbrella" mode (open protected). FDF is very fond of TC's commanding their wing men only by hand signals without using radio's and in CV9030 the "umbrella" mode is thus used as a main method for TC's (all situations except while under artillery fire). In Leo 2A4 you have to keep your hatch open IOT see or command you troops properly.

No. The hatch locks in place above the TC's head, it doesn't need to be closed and locked down. It can't swing from bumps as it needs to be unlocked with a manual lever to swing. It was routinely used in the "umbrella" mode to allow the TC to open it faster, at least in '05 and I see no reason why that practice would have been changed. You are right that it is not like the CV90 hatch in usefulness, but it nevertheless can be locked open above the TC's head.

News for you: it has been changed. And the reason is a injury risk. The problem is that you can't lock the TC hatch from movement up and down if it is in "umbrella mode. Yes, hatch can't swing sideways without opening the side locking lever. Also lift mechanism locking lever keeps hatch from moving in vertical direction in "Hatch closed" position and in "hatch locked open" position, but you can't lock it any "between" positions like "open protected". Check TDv 2350/033-10 and -20 for more info

I'll take your word for it, but I do find it strange. Certainly educated caution would have sufficed? It slows down opening the hatch considerably if you have to keep it closed all the way down, and if you must lock it open it obviously wont give protection...

The loader's hatch is far more dangerous than I can imagine the TC's hatch to be so long as the TC doesn't stick his fingers between the hatch when it's locked open.