I feel your pain; I just emerged from a terrible chemistry class. Here's your answers:
1) Remember that on a periodic table, the atomic mass (usually found at the bottom middle of the square) is the mass in grams of ONE MOLE of that element. For example, the mass in grams of one mole of hydrogen is 1.01 grams. In this problem, you don't have an element, but a compound (more than one element chemically connected). For these, just add all of the atomic masses together (39.1 + 54.9 + 16 + 16 + 16 + 16). There are four sixteens because there are four moles of oxygen (the O4). Adding these together gives you the atomic mass for one mole of KMnO4, 158 g. But you only want .57 mol, so multiply 158 g by 0.57 to get your answer, 90.06 g.
2) This is almost the same problem, but in reverse. You have the amount of grams in the compound, which we can convert to moles.
Atomic mass of Mg(NO3)2: 24.3 + 2(14 + 16 + 16 + 16) = 24.3 + 2(62) = 24.3 + 124 = 148.3 g
So, if you have 8.18 g of this compound, and a mole is 148.3 g, then the # of moles you have is 8.18/148.3, or .055 moles.
The number of moles in a compound are the same as the number of moles in each of the elements. Thus, there is .055 moles of Mg, etc. But, there are six times as many moles of O, so there are actually .33 moles of O.
3) I don't understand why you threw in 5H20. Because I don't want to give you the wrong answer, I'm not going to answer at all, unless you can clarify.