Yeah, your Luigi is pretty dang good xD. And yes, I've been playing Sheiks forever with Luigi.
Well-timed, weak Nairs work wonders in breaking combos.
Let's go over some basics. Sheiks are going to tilt you and jab you for some combo starters. Also, you might get Fair'd for damage racking. Dthrows won't be uncommon either, and Dsmashes can be used for spacing and damage.
One thing Sheiks will always do to you is keep you in the air. What they don't know is that Luigi has some of the best combo breaking moves. Now let's think for a second -what can Sheik REALLY do to beat a Luigi? Hmm... Floatiness = Fair won't send you that far until high %s. As a given, Uair won't kill until high %s as well. Dair? Yeah.. no. Bair? Only for edgeguarding. Nair? mmm not gonna work amazingly well on Luigi.
So WHAT does Sheik have to do? Sheik has to bring Luigi's percent up. How? Your opponent will struggle here. The only real way to bring the damage up is with Tilt combos and some aerials. Tilts will be easy. After being, most likely, Ftilted, let out an Nair ASAP. Chances are, some part of it will make contact. Depending on Sheik's %, follow-up smartly. If she's sent a little bit upward, then use the range of Luigi's Uair. If she is at high %, and in range for it, then Fair/Dair her. If you want to switch it up a little, feel free to Nair. I don't know what good it will do you though.
While Sheik struggles against you at low %s, you have your chances to combo her. Dsmashes are good at low %s into aerials. Also, at low %s, you can Uair, then fall into Fair/Dairs. Sometimes Nair works too. Shield is good too when Sheik isn't trying to grab. After you're hit, you can wavedash out of shield into a rab. Dthrow and Uthrow = t3h shiz on Sheik. At low % Dthrow > Dsmash. At a little higher % Uthrow > Aerial.
When Sheik is off the edge, hanging on the edge is a good place to be. I was once criticized for not standing on the edge to edgeguard Sheik. Hanging on the edge, however, is actually probably one of the best things to do against an off-stage Sheik. Many times, Sheik will try to go straight to the edge. This option is null because of your edgehog. Smarter Sheiks will go straight upwards, knowing that if you stay still their Vanish will hurt you unless you roll onto the stage. Then they fall onto the edge. Try and predict this; there aren't too many options in this situation. You can try just standing up on the stage (tap forward) and then aerial/Fsmash the falling Sheik. If Sheik is close enough to Vanish onto the stage, then roll onto the stage - but time it well! Sheik might still go for the edge, then you can still have your edgehog. Otherwise, you can grab/aerial/smash the Sheik when they get back on the stage. I prefer the grab and Dthrow into Dair; you won't have to worry as much about your spacing of aerials/smashes that way.
Surviving vs. Sheik: Don't worry too much about the Uairs, even when you're at a high %. If it hits, you'll probably die, but its hitbox is pretty small compared to how much Luigi can DI. DI to the side of her and aerial her - it's easier than you think. Then again, however, you'll get Fair'd off the stage every once in a while. You should follow standard procedure. Sheiks USUALLY won't edgehog you, but if they do, then do you best to get on the stage rather than the edge. Otherwise, Green Missile + Rising Tornadoes to get to the wall of the stage. UpB up the wall. That way, if Sheik Dtilts or Dsmashes you on the edge, you can tech it. Once you practice, teching is easy. Some Sheiks, however, like to wait for you to raise a little bit, then hit you with a further part of the Dsmash. For these, try and barely grab the edge with the UpB.
Once on the edge, you have many options. My preferred method of getting back on is ledgehopping into an aerial. If Sheik is shielding, try Dair > Nair in the one jump. They'll Shield grab the Dair right into your Nair. From there follow up with whatever combo works best in the situation.
Good Stages: I prefer platform stages like FoD and Dreamland. Sheik can easily get you with needles at FD. Stadium works well too for some variety. If you noticed, I counterpicked FoD against Tope's Sheik.
Scare the Sheik with combo breakers and finish her off. Good luck!
More questions welcome.