Your wallet file is not deleted when you uninstall electrum.
Also, the verification of the signature is not a mandatory step, but highly recommended to make sure you are not using a malicious version from a fake site.
You can find a tutorial on electrum.org.
Additionally, you should have a (non-digital) backup of your mnemonic code (the 12/24 words). You can't always rely on your AppData folder to be intact.
By "The long puzzle", do you mean the 12 words? It's called a "seed" and it's all you need to restore your wallet. You don't need the password when restoring from seed.
You can protect the mnemonic code with a passphrase. But that's not the password used for accessing the electrum wallet.