[RPG] Can monsters using innate spellcasting have their concentration broken

Yes, you can.

As stated in the spell Gaseous Form:

The creature is subject to the effects of wind, and it can’t enter water or other liquid.

Emphasis Mine, size doesn't matter in a gaseous form, as you will be subject to the effects of wind as any other gas would be

In addition to the effects noted, a gust of wind can do anything that a sudden blast of wind would be expected to do. It can create a stinging spray of sand or dust, fan a large fire, overturn delicate awnings or hangings, heel over a small boat, and blow gases or vapors to the edge of its range.

With Rules as Written, you can definitely cast Gust of Wind as a standard action, then using your move action to move into the gust, to carry you along.

If I was GM though, I would rule against this use, as the spell to me seems intended that a single blast of wind comes from the caster, moving away. The actual ability of the caster to get caught in the gust seems very ambiguous. It's your game though, and if you feel that the Vampire would be best served as a storyline villain, and not just a once off, that's something to discuss with the table. Just make sure you don't have anyone ready to snipe him off after the 60 ft. that he moves.

Scenario 1: The mount is charmed.

If the mount is charmed, then it will probably fall. From the section on Flying Movement in the Combat chapter (PHB 191):

If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.

Hypnotic Pattern sets affected creatures' movement to 0, so unless the mount can hover or is being held aloft by magic, it will fall.

It's unclear how long falling takes, so depending on your DM, the rider may or may not get a turn before they hit the ground. If they do, they can use their action to shake their mount out of its stupor as described in the spell. If not, their mount will fall and hit the ground.

If the mount can hover, or is held up by magic, then they still won't be able to do anything, but they won't fall, either.

Scenario 2: The rider is charmed.

If the rider is charmed, things get a little ambiguous. I won't post the entire text of the Controlling a Mount section here for copyright reasons, but suffice to say, it doesn't actually mention under what circumstances a rider can or can't control their mount.

If your DM decides that the rider can control their mount while incapacitated (seems unlikely, but you never know), then the rider will still lose their own turn but can (somehow) direct their mount on its turn.

If your DM decides that the rider can't control their mount while incapacitated, then the mount will act independently as described in the Controlling a Mount section. In this case, the rider will either be sitting there while the mount does whatever it wants, or possibly be thrown off. This is all pretty much up to your DM to decide.

Scenario 3: Both rider and mount are charmed.

As in scenario 1, if the mount can hover or is held up by magic, it won't fall, so rider and mount will stay exactly where they are. But if not, there's nothing either of them can do except fall helplessly to the ground.