A new coin usually does not have a wallet, and therefore you have to run a node on a separate computer.
Yes they do. "Wallet" really means the full node. Any other use of the term is only for marketing purposes to mislead.
If you don't have the keys it's not a wallet.
Just like "ASIC", the name has been corrupted by marketing and now many people call any mining appliance
an ASIC even when they don't use any mining ASIC chips.
Usually, in this option, a new wallet is created and a seed phrase for it, but I don’t know a single miner who can check the reliability of such a wallet generation. Usually everything works on trust and most projects do not survive.
Later, more accessible wallets appear, but the biggest profit can be obtained when the coin has only a node, and few miners mine it.