"paper wallet" and all cryptocurrency wallets(such as app wallet, hardware wallet and...) all of them just store public and private key into themselves(with different encryption ways)?
Essentially yes, all wallets are just different ways of storing private keys. Good ones such as paper wallets or hardware wallets are safest because they store them offline, mobile wallets and desktop wallets are not as safe because they are stored on devices which connect to the internet, and web wallets and some app wallets are the least safe because someone else stores them for you.
Which one of them has other mechanisms for protecting theft such as confirming transactions via email confirmation, SMS code, or...?
The only way to achieve this is by having a third party hold your coins, or using a multi-sig set up with a third party holding one of your private keys. Storing your coins entirely with a third party is not recommended for a wide variety of reasons (scams, hacks, poor security, poor privacy, being locked out of your accounts, KYC demands, etc.) so I wouldn't do this. The only other option I am aware of is setting up an Electrum wallet using their built in 2FA option. This sets up a 2-of-3 multi-sig with a provider called "Trusted Coin" who will co-sign every transaction you make after you provide a 2FA code from your mobile authenticator app. Note that this service charges their own fee, and because it is multi-sig, your transactions will be larger in size and therefore your network fee will also be larger.
