Yeah nintendo controllers are much better if you can get one. Nintendo stopped selling them recently, the only place I know of to get one new is ordering it from
play asia- the fact that nintendo stopped selling them is probably a sign that you won't be able to get them new much longer so if you want a new one you should probably get on that. If not you can always browse around for used ones at places like gamestop.
Other than that it comes down to practice, as does everything. Falco is one of the harder characters to wavedash with, just go in to training mode and spend some time with it.
As for the button inputs I don't think it really matters as long as you're doing what feels most natural for you. There's no apparent advantage of using L over R or vice versa. Y is closer to B than X, which matters for things like multishines (which isn't relevant to wavedashing I know, just throwing this out there) but people get to high skill levels using both buttons. PP jumps with X I believe so I think you can get good no matter how you use a controller as long as you do what feels best and practice with it.