I had one of the early keyhole operations for gall stone. They removed a stone the size of a small chicken egg and the bag out of a hole under my belly-button. There were three more holes for camera, tubes and- at a guess- inflation gas.
I was walking around in the foyer of the Oaks at Colchester within 36 hours. BUT - and yes, I'm shouting - I'd let it get bad before doing anything, and a couple of weeks before the op I was rushed to hospital as a blue light job. I felt like I'd got a spear in my belly that was coming out between my shoulder blades. I didn't know pain could reach those levels. I was stabilized with nothing but a drip and morphine for 3 days then sent home to wait for the surgery.
So, yes, get the bladder out of the way, you're so much better off without it.
Flying? No problem for me. Sensibly, about a month to make sure everything has settled down, but you'll have to see what the folks in the grey cube think. I had a class one for 15 years after that.