Does the private key have special mathematical properties?

[I read that there are private keys with lengths other than 256 bit. I only talk about 256 bit private keys here. Are other lengths even common and supported by popular wallets like Electrum?]

Let's say I generate a private key `privateKey0` using a key generator. Then I choose some number `n` ∈ [1, 10^10]. Given that `privateKey0`+`n` < 2^256, will `privateKey1` := `privateKey0`+`n` be a good private key?

A good private key of course needs to be valid and as strong as one generated by a private key generator. This does not take into account that one of them being made publicly available can lead to the other one being publicly available. However, using both productively must not weaken their secrecy for the second one to be considered a good private key. So let's assume that their corresponding public keys and the addresses for these are generated and both of them receive some amount of money.

I don't know anything about Bitcoin private keys (but am aware that an instance of the wallet import format is just the result of an easy-to-do operation on a big number as described here). Do they have to have special mathematical properties or can I just choose any number ∈ [2^255, 2^256 - 1] and use it as a private key?

Some websites say that you can just use a random number. Are there any disadvantages to this compared to using a key generator?