How can an output be spent while the transaction is in mempool
There is no rule requiring a transaction to have a confirmation before being spent.

Yes, the mempool matters. That's also how double-spends are detected. So it's crucial nodes check the mempool and not just previous blocks.
No it does not violate the rule, they are referring to double-spends being rejected if the SAME output exists in the mempool.
IDK, maybe ask the original poster what they meant?
Yes, no need for any confirmations to become an unspent transaction output.
Mempools from node-to-node are not always identical. For example, some nodes have different minimum relay rules that may include extremely low fee transactions. There is no central mempool.
Not really. I haven't done it recently but lets say you broadcast a transaction with a 0 fee. The transaction would likely get stuck in the mempool and not get confirmed. After 14 days if it doesn't get included in a block it &quot;expires&quot;. Once expired/dropped I could craft a new transaction using the same UTXOs that I used in the original.

Here's a related question that you will probably find interesting:
What happens to transactions in the mempool when their parent transaction expires?