>2. are there encryption methods that actually do not suffer from reductions in randomness or entropy etc when just simply running the algorithm on the encrypted output multiple times?

I think all block ciphers (e.g. AES) meet that definition. For AES, for a specific key, there's a 1-to-1 mapping of plaintexts to ciphertexts. It's impossible that running a plaintext through AES produces a ciphertext with less entropy, because if the ciphertext had less entropy, it would be impossible to decrypt to get back the plaintext, but AES always allows decryption.