Fishbowl

You have a glass sphere with a fish inside.  If the fish starts swimming, can it move the sphere?  Assume oxygen doesn't deplete and that the bowl is completely filled.

Specifically, yes.  It depends on how heavy the fish is, the coefficient of friction between the bowl and the surface it rests upon, how the fish swims, and if it hits the bowl itself.  Inertia will allow it to do something in certain conditions, and friction will allow that something to produce movement if it is low enough.

In one instance, if the bowl has a very low CoF below it, and the bowl is low wide, and the fish has a good mass and swims in a circle, the friction coupling of the water to the bowl should start the bowl spinning.  Placing such a bowl on a "lazy susan" would help.

The fish would move the globe. If the globe was filled to maximum pressure then the fish would not move, nor would the globe. And the fish would surely perish.

Conservation of momentum, right? If the fish starts moving to the right, the sphere would roll a bit to the left. At least until the fish hits the right side and then the sphere rolls towards the right.