There’s no need to remove the Omnipod; it’s waterproof. I’ve worn it swimming, in hot tubs, etc. As far as activities, you might have to watch it if someone is really roughhousing, but I think that’s the same with any pump; hit it in the right place, it’s going to get damaged or come off. Actually, one advantage of the Omnipod is that if it does get a direct hit hard enough to damage it, you only lose the pod, not the pump itself. Once you get used to wearing it, you tend to develop a sixth sense over where it is, and it doesn’t come off very often.
But to directly answer your question, no, you don’t take it off. Once you fill a pod with insulin and stick it on you, it’s there until it expires (3 days) or it comes off accidentally for whatever reason.
Some people wrap the pump with elastic tape or bands if they are worried that it will fall during activities. There’s all sorts of info out there about what to do when your kid is playing soccer/wresting/swimming; I don’t gather any pump makes it impossible to do anything a kid wants to do.
From reading posts from kids with Omnipods, the biggest problem I’ve seen is that some people (kids and adults both) have sensitive skin that causes problems with the adhesive, and/or if they get sweaty, the adhesive doesn’t work well. Once again, I gather that is similar with all pumps. When it comes right down to it, every one of them have a bit that sticks to you, and if the adhesive they use irritates you, that particular brand just might not work. I think these are edge cases, though, and most people don’t have issues.
As far as spots where you put it, the Omnipod can go just about anywhere you have a bit of fat on your body. The “official” spots are the top of your thighs, torso, and upper arms. In practice, I’ve seen people put it just about anywhere - calf, back, some women even wear it on the sides of their breasts (ouch!).
I don’t know a lot about tubed pumps, but I gather that most people put the insertion point somewhere on their lower torso or back. Though (I think) you can put it anywhere, you have to account for the fact that you have to thread the tubes through whatever clothes your wearing to the pump that most people wear on their waistband. I’ve heard women complain that wearing dresses is difficult with tubed pumps; with no opening at the waist, you have to be innovative as to how you thread the tubes.