No the private address or private key and the public key are derived using a mathematical formula. There are Services out there that attempt to give you a vanity address, but all they have done is generating a mass number of keys and parse the list that they've come up with for some common words names things like that.
Vanity address are different thing. It is possible to generate an address as long as you have a SHA256 hash, though the process will be complicated but it will work.
The private key is Created from the public key using math that same public cable always produce that same private key and vice versa.
No. You can get the public key from the private key but you cannot get the private key from the public key, at least without bruteforcing with little chance of success.