Looking at the way the algorithm works, is it safe to assume if no one is staking/mining the coin for longer than 10 minutes is it impossible to keep the blockchain alive?
For example, with master node setup if no one is staking/mining the coins for a while, then doesn't everything start to fall into "out of sync" because nothing has happened in X amount of time?
Yes that is what happens, if your coins is dependent on mining then there is no way that you can just stop mining the coin as that would crash your entire network. The entire blockchain basically runs on the processing power that is generated from all of the hardware from the people that are mining the coin. You would never be able to stop mining unless you want your coin to crash