Can someone clarify this for me?

I want to put Expert Handling on a ship.

If I fly the ship onto an asteroid I have to skip "Perform action" step and roll for any damage.
I think I should then be allowed to use Expert Handling and perform a free barrel roll (get a stress if I don't already have the barrel roll in action bar).

And since I barrel rolled off the asteroid I should also be able to shoot in that round.

According to the Rules Reference book pg 12,

FREE ACTIONS
A free action is an action granted to a ship through a card ability or other effect.

It does not count as the one standard action that a ship performs during the “Perform Action” step.
• A stressed ship cannot perform free actions.
• A ship can perform free actions even if it skipped its “Perform Action” step due to overlapping an
asteroid obstacle or another game effect.
• If a ship is specifically granted a free barrel roll action or a free boost action and cannot
complete the action in the desired direction, it may choose a different direction. It cannot
choose to perform a different action.

Also,

According to the FAQ pg 16

Q: If a ship is required to skip its “Perform Action” step (for example, if its final position overlapped another ship), is it
still allowed to perform free actions outside of the “Perform Action” step?
A: Yes.

Q: Can a ship that is overlapping an asteroid perform free actions?
A: Yes.

If I can't use Expert Handling as I described above then what constitutes a free action when I land on an asteroid?

No, you cannot.

The Action: of the card is the ability to trigger a free barrel roll. It's worded this way as previously (like Wave 1 or 2 previously) it didn't include the wording as it does now, so it was possible back then to actually use Expert Handling for a barrel roll, then trigger PtL for a second barrel roll. The Free Action of the EH card triggers from the EH card being active. You can't selectively pick and choose which brief portions of the card text that apply to a certain instance. The strength of EH is the ability to remove an enemy TL token from your ship after performing the BR

If another pilot/card ability gives you the opportunity to trigger a free action, like Squad Leader or Pilot Lando or Pilot Airen, then yes you could take a free action from being on an obstacle.

Any card with the ACTION: header require an action to "activate", after that, you get to do what the card text tells you. So if you are forced to skip your action step, you need some other way to trigger the card, like squad leader etc.

Thanks for the reply's.

So if I had Lando in his YT-1300 and Chewie in another YT-1300 (with the Expert Handling) and Chewie had landed on an asteroid which then causes him to skip his Perform Actions and roll for damage and cannot shoot, Lando when he makes his green maneuver could (if he's at range 1) have Chewie barrel roll off the asteroid and allowing him to shoot in the following attack phase.

Yes, as long as the template for the barrel roll did not touch the asteroid at all.