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?
If no one is mining, and this is very rare for any cryptocurrency, no transactions would be able to be verified, yet your balance would still show up assuming people are still running nodes.
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).
Can it be done? Yes. Is it useful? Not really.
If we all ran non-mining nodes, we could transmit transactions to the network, but they would never be mined into a block.
The network would still exist but in a "frozen" state. You still see your balance but cant do anything, no sending or receiving coins.