if you have your private keys you can create a new electrum wallet with them. Here are instructions for how to do that on a PC:
https://bitcoinelectrum.com/importing-your-private-keys-into-electrum/Here are answers to some of your other questions:
Each address has a different key pair behind it. Private and public keys. The public key is derived from the private key and is mathematically related to it. The address is derived from the public key.
Private keys are not the same as public keys. Private keys are secret while public keys are not. Knowledge of the private key allows a person to spend money sent to the corresponding address.
You can see what private keys look like here:
https://en.bitcoin.it/wiki/Private_key#Base58_Wallet_Import_format . Addresses are shorter and begin with 1 or 3
https://en.bitcoin.it/wiki/Address. Public keys are large hexadecimal strings ( case insensitive 0-9,a-f).
Watch-only wallets don't have private keys in them so you can't spend from watch-only wallets.
All private keys in an electrum wallet are derived from the seed. Therefore backing up the seed allows you to recreate your whole wallet.