Can the Monero network exist without miners? Can we just run monerod on home computers? Must some of us use the
start_mining command (or a similar command in other mining software) to support the network?
No it cannot exist without miners as no transactions would be verified and included in blocks, and no new coins would be created. In other words, the blockchain is created by miners. If there are no miners, there is no blockchain (or no continuation of the blockchain if you are starting with one).