How private key is generated from password in Bitcoin-clients We all know that private address is a 256bit number. But is the process of converting your password to this 256bit number standard across different clients?

I can imagine that a "strong random 40 character password" after transformation/hashing et c might become the stupid 0...000000123 or something like this :)

If this process is standard, does this mean, that backup of wallet.dat is not required, you should remember only the password?
It's not standard across all clients. You mentioned wallet.dat, which I associate with the Bitcoin Core client; that does not use a password to generate private keys; it generates each set of private keys randomly when a new supply of them is needed for transactions.

Some other kinds of wallets, however, are known as HD, or hierarchical deterministic, wallets. When they are initialized they generate a random master key that they use as a basis for generating all of the wallet's subsequent private keys in a deterministic (non-random) way. And many HD wallets, the very first time they're used, display a sequence of natural language words known as a recovery seed. You write down the recovery seed word list on paper and keep it securely secret. If the wallet is lost or destroyed you can input the words into another one of the same type of wallet to re-generate your private keys and hence your coin balance. If you're interested in the technical details, look at the specifications for BIP32 (HD wallets) and BIP39 (recovery seeds).

An HD wallet which uses a recovery seed may allow the user to specify a passphrase, typically intended to be memorized rather than written down, which is used along with the recovery seed to (re-)generate the private keys. For the ones that I've encountered a passphrase is optional.