I really don't think the projectiles are too much of a factor in this matchup, except for a single regard: if Falco can hit Sheik during a distant horizontal recovery, it stops her momentum and can easily make for a gimp. The trouble is that this method is really the only gimp advantage Falco has on a smart Sheik. Sure he still has the dair spike, but any smart Sheik will see this coming and punish it with a vanish (unless you are very sneaky/quick) As far as needles, these will probably only be used on occasion against Falco, when he is offstage or using phantasm or his AAA flurry thing.
It can also be added that Falco's chaingrab and Sheik's ftilt lock are almost equal in different respects, as each builds up a similar amount of damage and can lead into decent finishers.
On the ground, Falco's fsmash definitely has a range and speed advantage over many things that Sheik can dish out, unless Sheik can predict it and get a chain tipper in before the attack completes. This is not something to expect, though, as most Sheiks will not choose this route and would rather just dodge and then aerial punish.
Aerially, Sheik's bair keeps Falco in line, and if she sees that you aren't dairing, she can even get in quick uairs for a juggle or two.
The main hindrance for Falco will be once he is offstage. Her fair will have no problem keeping Falco under the stage, from which his only recovery option is his UpB. Unfortunately, this is ***** by the chain dangle from on stage. In addition, offstage Sheik can hinder this with a quick tether recovery or a slightly faster vanish if she is on the inside.
Overall, Falco will have the easier kill moves with a quick Bair, Fsmash, or maybe Dsmash, but Sheik will get the easier gimps and be able to control the speed of the match. She also has the fallback option of turning to Zelda to avoid the CG, get the kill, and generally improve survival times - provided the transform is done safely.
I'd call it even for the hell of it, but if there is an advantage it will go to Sheik for her gimps on Falco.