When you enable record padding for a client, does a server need a specific option to support this?
It shouldn't; record padding is a part of the RFC which is not listed as optional; I don't see anyone could claim to support TLS 1.3 without understanding record padding on decryption.
If I would choose a factor of 2 it would make the record (=packet/payload length???) twice the original size. Is that correct?
No; if you set a factor of 2, then the encrypted record size would always be even (that is, if it would have been odd, a padding byte would be added).
In general, a factor of $x$ would increase the size of the encrypted record from ...