Firstly can it effectively be ordered around with a free action, taking actions on its own turn? I assume yes.
Yes, you can speak as a Free Action (Other Activity on Your Turn)
You can communicate however you are able, through brief utterances and gestures, as you take your turn.
And the Planar Bind makes it obey your commands
A bound creature must follow your instructions to the best of its ability. You might command the creature to accompany you on an adventure, to guard a location, or to deliver a message. The creature obeys the letter of your instructions, but if the creature is hostile to you, it strives to twist your words to achieve its own objectives. If the creature carries out your instructions completely before the spell ends, it travels to you to report this fact if you are on the same plane of existence. If you are on a different plane of existence, it returns to the place where you bound it and remains there until the spell ends.
Monsters (can) do the same as PCs, as long as they fit the requirements and nothing on their statblocks say otherwise
This is the general answer for your questions 2 to 4, except for the Multiattack thing. I'll now handle each one individually.
Secondly, can it grapple? Again I assume yes as there is nothing to say otherwise? Since it has multi attack can it grapple two targets?
This is actually two questions: yes, monsters can grapple, as long as they meet the requirements from the grappling action - being the right size and having a free hand. Air elementals seem to have hands, so they should be able to. That depends on your DM interpretation though.
No, monster with multiattack can't grapple more than once in the same turn. TL;DR: Multiattack is not the Attack Action, but an action of its own. Monsters don't make multiple attacks as a part of the Attack Action, which is the requirement for grappling, rather, they take the Multiattack action. Grappling:
If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.
Note that you can, however, grapple more than one target (or two targets), which is what you actually asked - but that is going to take two turns and has nothing to do with Multiattack.
Does trying to move with two grappled target reduce it's move distance twice? 22.5ft fly speed?
The Air Elemental has nothing about special grappling (i.e. no specific rule for it), so we use the general rule, which states
Moving a Grappled Creature: When you move, you can drag or carry the Grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
It's not clear if the speed should be halved twice when grappling two targets. The grappling seems to assume you are grappling one creature. By that interpretation, yes, it would cut it to 22.5 ft. Another interpretation is that even if it applies to each grappled creature individually, no, it does not stack. IMO, the rules are not clear enough for this case (and I didn't find anything on Crawford here), so I'm throwing it to up to the DM for now.
We already have this question here. Note that there are two contesting answers - and personally I don't think the accepted one has enough textual evidence for it to be RAW right.
Thirdly, can it whirlwind as an action while grappling two targets. If so, what happens to them?
Whirlwind doesn't state it requires a free hand, so technically it can. YMMV though, depending on your DM. Either way, releasing the grappled targets is a free action, so even if you can't, the Air Elemental can just release them and use the Whirlwind anyway.
"What happens to them?" is vague - but they will be released from the Grappling and Whirlwind takes effect anyway. The reason they are released from the grappling is:
is flung up 20 feet away from the elemental in a random direction and knocked prone.
which will put the enemies out of the Air Elemental's reach.
Lastly, what happens when it carries a willing creature? Is its movement speed halved as usual, does encumbrance come into play?
Again, there is no specific rule stated in the Air Elemental, so we are using the general rules from PHB.
Crawford says that carrying friendly PCs applies the Carrying and Lifting rules from p. 176 and that there is no distinction made between flying and walking.
See "Lifting and Carrying" in the Player's Handbook (p. 176) for rules on carrying capacity.
The carrying capacity rules make no distinction between walking and flying creatures.
So, no, speed is not halved, although it is reduced to 5ft if (and only if)
While pushing or
dragging weight in excess of your carrying capacity,
your speed drops to 5 feet.
From comments, that carrying capacity is 420 lb.
Encumbrance is a variant rule, so it depends on your DM using it or not. Again, there is nothing there about halving the speed, though. Rather, we have fixed amounts of movement speed loss.