Prop Bets - Last Longer - If I buy staking in a player, does that include any Prop Bets they may enter?

Prop Bets are entirely separate from staking. It's impossible to buy a share of someone in a Prop Bet.

All staking transactions refer to the player's performance in the event only. Payouts depend on that player winning cash in the event, either via finishing in the paid places, collecting bounties or winning a Bounty Jackpot.