2) Suppose I notice the following prices: European put with exercise price $50 and one month to maturity is selling for $3

All else being equal, the longer the maturity of the option, higher should be its price and vice versa.

Here, there is an arbitrage opportunity because the put option with two months to maturity is selling for a lower price than the put option with just one month to expiry for the same strike price.

We can buy the two-month put option and sell the one-month put option.

We receive a credit of 3.20 - 3.10 = $0.10

This is the arbitrage profit. After one-month once the near-month put option expires we can sell the far-month (two-month) put option to realize the gain.