I'm currently in a campaign that has grown to a ridiculous size (9 players, although one of them doesn't really show up any more). Because of some half-joking actions of the only Dwarf in the party early in the campaign, we all have dragons as mounts, now, including my Warden.
The biggest pitfall to be wary of is to not turn mounts into another party member. If the player wants the mount to act in combat, he must use up his own actions to command it. Assuming you're already doing that, the mount becomes essentially another magic item, which offers alternative attack(s) (which, trust me, are very quickly outclassed by what the player has access to), alternative movement modes, and can become a liability (dominate, prone-in-flight, no-hover flight, etc.).
Note that, RAW, the mount starts getting its own set of actions the turn after the rider dismounts. Personally, I recommend against letting the mount be another party member; even if the mount's attack bonus is too small to be useful, setting up flanking is extremely effective. In the above campaign, dragons that have been dismounted back away from the combat until their rider wants to re-mount (or the rider wants to use actions to command the dragon while dismounted), but that's house-ruling by the GM.
Now, to answer specific questions:
What are particular dis-advantages of using a mount, or a flying mount?
The mount doesn't level up with the party, and so the mount will quickly fall behind. If the rider doesn't have the Mounted Combat feat, it's even worse, as the mount will have a -2 attack penalty. Eventually, the mount's attack powers won't be worth using, and the mount will be hit by just about everything except a natural 1. Mounting or dismounting requires a move action, and proning the mount will dismount the rider (and prone him... and standing up requires a move action), which slows down the character a lot. Consider: I prone a mount and its rider. On the next turn, the rider stands with a move action, and converts his standard to a move to make the mount stand... then the turn after that he uses a move action to re-mount. With a single attack, I've consumed three of the target's actions, and he hasn't left his square.
For flying mounts, you add the problem of falling when they get knocked prone (although you need to be higher than the mount's move speed to do damage). If the mount doesn't have hover, they'll also fall (and get knocked prone if they fall from a high enough altitude) if they don't move 2 squares on their turn, and they can't shift. This provides a lot of opportunity attacks.
Can a player with a griffin mount just always fly or when would the mount be exhausted?
Yes, the player can always fly, assuming there's room to do so. In the campaign I described above, the GM alternately builds encounters in the open where we can use our dragons and in enclosed spaces where the dragons can't fit, forcing us to hoof it.
Can you fly upwards with a griffon the same speed/distance as dropping down?
Yes. The mount's speed is the same in all directions. (Although I suppose you could move a larger number of spaces by falling, that comes with prone and damage.)
If the mount has a speed of lets say 12, does the speed decrease when a character is mounted?
I don't know of any creature whose speed is reduced by having a rider. I suppose it's possible for such a creature to exist, though.
When and how is damage dealt to the mount, when the character is mounted?
When an enemy uses a single-target attack, the attacker can target either mount or rider. It doesn't matter what square the rider is on; if the mount can be targeted, so can the rider. Area and close attacks which include the mount include both mount and rider. However, the two are separate entities. If the mount is dealt damage, the rider is fine. If the rider is slowed, the mount isn't.
Can a mount be revived?
Sure, why wouldn't the mount be able to be revived?
I feel that it's worth mentioning that neither the Griffon, Bronze Griffon, nor Rimefire Griffon have the hover quality. That means the griffon cannot shift in the air, and the giffon must move at least two squares or fall to the ground. Since you can't shift and you must move, the rider is likely going to be provoking a lot of OAs, especially a melee character on a griffon. Also, an immobilized/restrained/stunned/dominated griffon is going to fall like a rock.