If somebody is engaged (close combat). and I have a taxi right next to my engaged character. Do I have to break away with my engaged character in order to use the taxi?

Depends on how they are positioned.

E
GF

E=Enemy
G=Grounded Friendly
F=Flying Friendly

In this case, unless your flyer had Phasing or a similar ability, the flyer would have to roll for breakaway to carry off the grounded figure because they too are adjacent to an enemy figure.

But if you do this...

E
G
F

The Flyer is now free to pick up the grounded figure and fly away without having to roll for breakaway.

So the taxi'd individual can be dragged out of contact without having to roll for break away? Huh!

I guess that is what all those DC figures wear capes for... to give the taxis something to hold onto as they drag them away...

Yep.

You aren't giving an action to the carried character to be carried so no breakaway roll needed.

In that Scenario:
E
G
F

Can E hit F with a ranged attack? I'm still a bit confused about whether grounded characters (like G in this case) block LOF to flying characters (F).

No E cannot hit F. They do block LOF, the only exception to this is if there is a different elevation involved (if F were soaring, or E were on a high-rise...but then the break-away question wouldn't be posed...)

This should solve your problem...

Any character that is not soaring or elevated is grounded. So a non-elevated hovering character is grounded.

This means every character in your diagram, regardless of whether they are a hovering flyer or a character without flight, is grounded. Meaning G blocks LOF between E and F.

That does indeed solve my problem. Thanks!