As for beating out a Sheik recovering low, this is very easy: WD onto ledge, wait for her to land on stage, punish. If Sheik is slightly farther away and can make it on stage, use PKFs to make sure they stun her just enough so she will fall a slight bit, and be forced to UpB on stage. If you can catch her as she's jumping right away onto stage, try rising Bair to spike her or Fair to swat her away.
tl;dr Avoid getting grabbed and watch for her tilts. Don't be reckless at any time. Capitalize as often as possible. Sheik doesn't have much end lag to punish most of her attacks. Watch for any openings that you can find. Adapt as well as possible and be sure to have keen spacing. Play this MU similar to vs Marth.
A Brief Summary of the Sheik v Lucas MU
Neutral: Avoid needles and grabs. She can chain grab to mid % which is where combos become deadly on Lucas and may lead to gimps. Use platforms to maneuver needles and avoid Final Destination most of all. Make sure to not get caught in the air as she has many anti-air counters with tilts and great aerials. Approach very cautiously. PKF can only be stopped by her needles, Nair and I believe Ftilt. There might be a few other moves that counter PKF, but these 2-3 are the main moves that Sheik players will use. Needles always beat PKF and is advantage to Sheik because both characters have similar end lag on these moves, but one needle is all it takes to beat PKF, the rest is extra damage on Lucas. Nair only works if she's landing, at which point you can rush Sheik down while she's landing and punish her with a grab or attack. If Sheik does a rising Nair, trap her landing. Note, punishing Sheik's Nair should only be done if you are within 1-3 character lengths and just outside her Nair range. Do not try to punish more than 3 character lengths away.
Combos: Pretty free on Sheik with her weight and fall speed. Hard to start up as she's resilient to getting hit / grabbed. Once you land a hit it's hard for Sheik to do much.
Ledge guards when Lucas is recovering: Avoid Needles and rising aerials from Sheik. Make sure to mixup recoveries very well and do not be afraid to eat a little damage by going high. She is very proficient at gimps and killing Lucas at low heights. Be wary of using your double jump off stage. Once you're hit out of it, recovering becomes infinitely harder, especially in this MU
Ledge guards when Sheik is recovering: As mentioned above for recoveries going low. When going high, there is a certain range on Sheik where she has no vertical hitbox diagonally below her that a rising Fair/Uair/Bair can hit Sheik that she can't hit you. If she does not have a double jump, punish the landings with Utilt/Ftilt at low % Upsmash/Fsmash (especially OU smashes). PKF to catch the double jump or safely space back safely shooting PKF and bat away her high recovery with Fairs and such.