Can you carry when doing a HSS attack option one?

Yes. You just have to drop them off before you attack.

Not true.

A flyer with HSS can carry a grounded figure, but during the HSS attack the carried figure is considered to be off the board. So if say KC Superman (on his HSS clicks) carries V Psylocke and makes a ranges attack as part of his HSS, he does not get the Enhancement bonus to damage. Only at the end of the movement is the carried figure set back down on the battle map.

Normally, the order of things during a HSS attack by a flyer is --

1) Pick up adjacent friendly figure
2) Move (although your movement might be 0 squares)
3) Attack (from squares where the flyer and the passenger can legally occupy)
4) Finish movement (once again, your movement may be 0 squares here too)
5) Put down friendly figure in an adjacent square

What about grounded transporters using the transporter attack? For example if Ghost Rider is carrying a DEO Agent, can he benefit from her police team ability when making an attack?

Nope. The transporter move-attack-move really is "HSS lite". Any passengers are considered off the map at the time of the attack, just like HSS. Basically, the only difference is the transporter MAM attack and HSS is the MAM attack at -2 AV.