I only have $5. My opponent lands on an un-owned property and put it up for auction. Can my opponent make a starting bid of $10 on this auction even though that is more money than I have?
This is definitely allowed, and a good tactic to leverage a cash advantage.
There is no minimum or maximum price for an auction. The rules state:
Bidding may start at any price.
Monopoly auctions do not have "the bidding will start at $10". They just have "bidding is open". Once the auction begins, a bid by any player sets the minimum price for the auction at that bid. Your opponent could easily bid $10 on the property, forcing you to either let them have it for $10 or bid more than you have in cash. You could bid first, say starting the bid at $5, but that doesn't meaningfully change the situation, as they can just respond by out-bidding you at $10.
You may bid more money than you have in cash, but you are still obligated to pay it if you win the auction. This may be a reasonable play if you can afford it by mortgaging property and/or selling houses, especially to drive up the price for your opponent. But be careful not to bid more than you can pay after mortgages and sales, as if you win the auction and can't pay your bid, you go bankrupt. In particular, if you bid more than you can afford, you opponents can all just let you win the bid, at which point you go bankrupt and the property you just bought would go up for auction again (in addition to all your other properties).