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I can give you a definitive answer to the first part of your question. Yes, formula units do mean the same thing as atoms. As for the second question, I would assume that formula units do not change for dissolved ionic compounds due to the fact that atoms are neither being created nor destroyed during this process. I could be wrong about this.
In terms of the problems we are solving, I think they are the same since they are considered the same in the textbook questions. One mole is 6.022X10^23 of anything, so in order to calculate, just use Avogadro's number to solve for either atoms or formula units.
Whenever a question asks you to find the formula units, it's simply asking you to find the number of atoms. So, if the question gives you the number of grams, divide it by the molar mass in order to figure out the number of moles. Then use Avogadro's number to find the number of atoms, which is the number of formula units.
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