RSA / EC Key Pair Generator creates a key pair in the page with the Web Crypto API and exports it as PEM: the public key in SPKI form and the private key in PKCS#8 form.
The keys are generated locally and are not sent anywhere. The private key is shown once, so save it before you leave.
Elliptic-curve keys are smaller and faster for the same security level and are a good default for signing. RSA is still widely required for compatibility. P-256 or RSA 3072 are reasonable choices.
The public key is SPKI (BEGIN PUBLIC KEY) and the private key is unencrypted PKCS#8 (BEGIN PRIVATE KEY). Most tools and libraries read both.
No. It is exported in the clear, so store it somewhere safe and consider wrapping it with a passphrase using your own tooling.
No. Generation and export both run locally with Web Crypto.