1. Can Supreme HQs deploy one hex?
Yes, I believe so.
2. Rule 5.62 states that a SHQ can Mobilize using normal or strategic movement. If it uses strategic movement in dry conditions, can it actually move up to 20 hexes by rail (because it has a SPEED of 2)?
Well, a unit's speed doesn't affect how far it can move by rail. All units move 10 rail hexes per supreme move; 5.62 is saying that a SHQ can Mobilize by rail (making one 10-hex rail move) without spending a supreme move. If you want to move 20 hexes (which would normally take two supreme moves), then you would save one of your supreme moves for that.
(That's what's meant by the last sentence of 5.62: if you activate your SHQ at 3CV, you've got six supreme moves; if you use five of them on other units, then you've got one left over which you can use while mobilizing your SHQ.)
3. Rule 8.1. states that hex control is determined at the beginning of each phase and remains unchanged throughout this phase. So a passive unit that becomes engaged during an enemy movement phase still makes all hexes within its ZOC enemy hexes to the active player? In our specific case: The passive Russian unit's ZOC blocked a vital rail line and thus made an important German Rail move impossible. The German then sent units into the Russian hex to negate its ZOC. This was not correct, was it?
Right, the hex would continue to be controlled by the Russians for the rest of that movement phase. (And beyond, if no German unit had occupied or exerted a ZOC into the hex--remember that German units moving through the hex to engage the Russian unit don't change control of that hex.)