I'd like to understand how exactly private keys are converted into pub keys and addresses
You want to understand how Point Multiplication in Elliptic Curve Cryptography works for the process of computing public keys from private keys. This link has some easy-to-understand content:
https://blog.cloudflare.com/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/From public key to address, the links you shared are already explaining that.
but for all elements (well, except testnet)
Considering keys and addresses, the only difference between MainNet and TestNet is the version byte in Base58 encoding of P2PKH and P2SH addresses and the human readable part of the Bech32 encoding of P2WPKH and P2WSH addresses. The rest are the same.