For 1, when I learned multiple integration I found useful to think in a way that the coordinate system gives you parameters that describes the solid, or the region you are dealing with. Although this could mean the same as we always do, maybe it helps you to understand why the integrals vary from 0 to , for example.
So imagine the solid's shadow in the xy plane: you will get a big circunference with radius 2 formed by the intersection of the two paraboloids. (just plug one z into the other one, then you will find the equation of that circunference I'm telling you). We must have - in a heuristics way, we will sum a series of solids each one with different radius, but similiar in the shape.
For the lower and upper bound of z (the height) you should see that gives you the lowest heigth and gives you the highest height, but, every size will depends on the value of r so we must write the two equations in polar form and we will use the r as the limits for the integral with respect of z, or the height. If , then by doing and we obtain and for the other , we must have by the same substitution.
Finally, fix a radius, and fix a maximum and minum height - you should have a line describing the contour of the solid. If you rotate this contour by 360 degrees, or , you will obtain a surface of revolution. Thats why the angular parameter of the cylindrical coordinates should vary from 0 to .
So mix all these together and think a little about it then you should see that this makes sense in a weird way - we will find the volume of the solid adding all these "shells" that grow in size as we change r. But if you see, we must integrate two times to get a "thin plate", then the rotation of this will give us the required solid.
So, we must write the integral taking care that the order of integration is correct, because we see that the height depends on r, so we must integrate first the height then the r:
Noticing that is the area element in the xy plane which is .
I hope this helps you in this exercise and in the others and I apologize for any mistake in the english.