No. Online play is really, really bad for helping you improve.
I disagree because it allows you to fight a wide variety of players in a relatively short amount of time. Because of Brawl's online, I actually have competition for the first time in almost 4 years and I have grown tremendously because of it.
Yes. Lag trains your brain to think slower, and therefore not only is there a shallower skill cap for online play, but it actually hurts your ability to play in a faster-paced and more precise offline setting.
That's the thing though, you talk as if there's a continously lapse of lag for everyone and that's definately not the case. In the majority of matches that I'm in, I can still powershield and use boost smashes. It all has to do with your internet connection.
Just watch matches online between top-ranked ladder players and offline tourney matches between pros. There is absolutely no comparison. And while offline pros can do well online, top-ranked online players almost never place in offline tourneys.
Just wanted to throw that out there. If you want to get good, even with a shallow game like Brawl, do NOT PLAY ONLINE.