Nice alliance auction in 3 players!!! (Alliance A 2 players and alliance B 1 player).
Yeah, I like this game with three, partly because the alliance bidding is so brutal.
We played that buying a "Senate Vote" card counted as action "Buy 1 Conquest Card".
Yes, Senate vote cards count as Conquest Cards. (On page 6, where it's describing Conquest Cards, Senate vote cards are included.)
Can I win the senate vote using 1 single card and get it back in hand for having more Influence in Italy?
we played YES.
Yes, I don't see anything prohibiting this. Note that Miss Manners frowns on laughing at your friends for more than a minute or two when you do this.