Hi
it was a good attempt, you figured out most of it. I will answer all your questions, but first let me mark the main clause in the first sentence which evidently gave you the most difficulties.
οὔτε ἀγαθὸς ψυχῇ οὔτε σώφρων ὃς ἄν ποτ' ἔρωτί τε καὶ σώματος κάλλει δουλεύων βούληται τὰ τῆς πόλεως πράττειν
With red you have the main clause, with blue the subordinate clause. We only have 2 clauses in this sentence. The verb of the first sentence you'll have to supply: "ἐστί" = "he is". The verb of the second clause is of course "βούληται". Now...
1: There's actually 2 ways you could approach this. You can either say that the whole subordinate clause is a subject to the verb of the main clause (ἐστί). Or you can be more precise (but also a bit more complicated), and supply "οὗτος" = he. Then of course the whole subordinate clause would be modifying our subject οὖτος. This might be a bit tricky, so please if you don't understand exactly what I mean let me know and I'll rephrase.
Now let me provide the translation:
Neither good-hearted nor prudent (is*) he who, while being a slave to love and the beauty of the body, would ever want to be involved in civic affairs.
*which we have to supply
So, the subject of our main clauses verb is the whole subordinate clause. Otherwise, we could have also supplied one more "he":
He is neither good-hearted nor prudent, who, while being a slave...
Then the secondary clause would be modifying "he" (οὗτος)
2: As you will have noticed from my translation, ἀγαθός ψυχῇ means good hearted. Literally "good with respect to the soul".
3: Let me clarify that ἄρχοντος here is a participle of the verb ἄρχω, not the genitive of the noun ἄρχων (of course the noun is derived from the very same verb and it's participle, but still, the role of the word "ἄρχοντος" in our clause is that of a participle). τούτου modifies this participle, but refers to this man. So in essence it both modifies ἄρχοντος and means "this man". And in fact, ἄρχοντος is a participle that expresses possibility (a conditional participle, if you like), which is why your translation of that clause is perfect.
One more thing:
πῶς γὰρ ἂν οὗτος ἄρχοι = for, how could he rule
Given that it's the construction ἄν + optative we've encountered before, I'm sure you'll understand why it means that, and not "how does this man rule".