It seems you are confused with the definition of moles and its relationship to molecules. Saying I have one mole of water is similar to saying I have 1 kg of water. Both have a set quantity. But what is the difference? Grams is a measurement of weight. Moles is a measurement of number of molecules. One mole ALWAYS contains 6.022 *10^23 molecules of any given substance. One molecule may be heavy or light which is why one mole of iodine is very heavy compared to one mole of hydrogen gas.
I will answer a few of your questions, then I hope you understand the trend enough to answer the rest of them.
Remember: 1 mole = 6.022 *10^23 molecules.
So for a, 2 moles = 6.022*10^23 * 2 = 1.2044283 × 10^ 24 molecules.
b. 6.022 * 10^23 * 4.5 = 2.70996367 × 10^24 molecules
On the other hand, when you have the number of molecules, you essentially do the opposite of what I did above. 3.01 * 10^23 molecules / 6.022*10^23 molecules = ~0.5 moles
1.000 * 10^23 molecules / 6.022*10^23 molecules = 0.166057788 = ~ 0.1661 moles.
When you refer to moles of something, it would be like saying how many dozen of an item are you talking about. Instead of saying how many groups of 12 do you have of a certain item (in this case how many dozens), moles would be saying how many groups of 6.02 x 10^23. of a certain item do you have.
Example: If you have 12 eggs, you have a dozen eggs. If you have 42 loaves of bread, you have 3.5 dozen loaves of bread. If you have 9 dozen of doughnuts, you have 108 doughnuts.
If you had 1 mole of H2 molecules, you would have 6.02 x 10^23 molecules of H2. If you had 9.03 x 10^23 molecules of water, you would have 9.03 x 10^23 / 6.02 x 10^23 = 1.5 moles of water molecules.
If they ask for number of molecules, your answer will be in number of molecules. If it is ions, it would be in number of ions. If it is number of molecules, it would be number of molecules.
So to answer your questions:
a) 2.00 mol Fe3+)(6.02 x 10^23) = 1.204 x 10^24 Fe+ ions
b) 4.5 mole BCl3 = (4.5)(6.02 x 10^23)2.709 x 10^24 molecules of BCl3
c) 0.25 moles K+ = (0.25)(6.02 x 10^23) = 1.505 x 10^23 molecules of K+
d) 6.022 moles O2 = (6.022)(6.02x 10^23) = 3.625 x 10^24 molecules of O2
This part is like saying how many dozen do you have
a) (3.01 x 10^23 molecules H2O) / (6.02 x 10^23) = 0.50 moles H2O molecules
b) (1.000 * 10^23 atoms C)/(6.02 x 10^23) =0.1661 moles C atoms
c) (5.610 x 10^22 ions Na+)/(6.02 x 10^23)= 0.0932 moles Na+ ions
One has to understand what a mole actually is. It is just a number chemists use to express weight. This many atoms of any element equals its atomic weight in the periodic table. Iron (Fe) weighs 55.8 grams/mole. 2 moles * 55.8 grams/mole = 111.6 grams. However, the exercise forces the student to actually use the value of the number itself. This is 6.02 x 10^23. Just multiply this number by the moles given to get the number of atoms, ions, etc. If given a number of atoms, ions or elephants, just divide the number given by 6.02 x 10^23 to get the number of moles.
/////////// FYI : this is absolutely the worst way to teach chemistry. No (sane) chemist ever counts individual atoms. Chemistry is neither physics nor geometry. PI is just a number but the value is very important. The actual value of the number "mole" is not important what so ever.