You can be invincible throughout the entire attack portion of it, but you won't be able to regrab the ledge before your invincibility runs out and will be vulnerable for at least 6 frames. That would be pretty beast if it were possible though.
I generally do the ledge regrab u-airs and ledge hop onto the stage u-airs differently from each other.
To regrab I usually do a very quick 1/4 circle away to up on the control and press A right as it reaches the up position. Since I generally only do this instead of landing on the stage with it for the extra range, doing it this way makes you doublejump away from the ledge slightly giving it significantly more range than most methods. To get the absolute max range on it I'll also sometimes hit away, jump with Y using my index finger, c-stick the u-air, then start holding back towards the edge with the control stick, which is kind of like how I do u-airs onto the stage.
To land on the stage I do it similarly to what Frozenserpent said. I do a quick 1/4 circle from down to towards the stage. I only tilt down though before moving left/right to prevent fastfalling, then press Y once the analog gets to left/right, then press up on the c-stick right after that, and then l-cancel. I double jump stall using my thumb on the X button to jump, then when I want to ledgehop an u-air I move my thumb underneath the c-stick and the inside of the first digit on my index finger ontop of the Y button. Though the transition might be awkward for most people, it's 2nd nature to me from using Peach and floatcancel u-air comboing and such. I find this way to be SO easy if you can switch to it without much thought.
I never really liked using the c-stick to let go of the edge personally though. Didn't like the idea of falling with a b-air if I were to hit it a little off.
Something most people don't realize is that what's most important is what direction you are holding when the jump is initiated, and much less so what you hold afterwards. The direction you hold on the analog the instant you begin to move from the jump determines your main jump trajectory, and then from there you can move around slightly using air control on the analog. For example, you can let go of the edge, jump while holding towards the stage, immediately let go of the analog completely the moment the jump starts, and the jump trajectory will still carry you well over the edge anyway. The same principle applies to ground jumps, only it takes the direction right as you leave the ground after the startup lag not when you press the jump button (since an air jump happens immediately and ground jumps do not).