1. Is your multi-sig backup used only in case of emergency if you lose your (hardware) wallet, or you forget the PIN number of your wallet?
A multisig backup is like a singlesig backup. You can use it for various reasons, but most importantly to never lose your funds.
Imagine having one wallet (A). You would keep a backup of the seed phrase somewhere. Correct?
Now imaging having one more wallet (B). You will also keep a backup of the seed phrase somewhere.
Those two wallets can construct a multisig wallet where one (1-of-2) or both of them (2-of-2) are needed to sign a transaction. In fact, I don't think a 1-of-2 is useful, because you can use a singlesig wallet instead.
2. Or do you always require third party permission(s) in order to sign for every normal day to day transactions you want to make?
There are basically 2 options:
1. you control all the cosigners (keys).
2. you control most of the cosigners and there is also one entity (company) holding one of your keys. This is called collaborative custody.
I always use option (1).