How to recognize Authenticated Encryption with Associated Data ciphersuites
Can you help me identify the ciphersuites that provide Authenticate
  Encryption with Associated Data (AEAD) from those that do not?


All which explicitely name gcm or chacha20_poly1305 are AEAD ciphersuites. The others use the TLS default which is MAC-then-encrypt with CBC.


  Is it always the case that AE ciphersuites must be also Forward
  Secrecy ciphersuites?


There's no technical reason that forces this to be the case. In fact RFC 5487 defines TLS_PSK_WITH_AES_128_GCM_SHA256 which uses AES-GCM (AEAD) but doesn't offer forward-secrecy. And if you don't accept a PSK suite as an answer here, RFC 5288 defines TLS_RSA_WITH_AES_128_GCM_SHA256, which uses RSA key-transport (and thus offers no forward secrecy).

However, AEAD is what some people call "modern crypto". And as such it is usually paired with other "modern crypto", which does usually mean that you prefer to use forward-secrecy with AEAD suites and only have the non-forward secrecy ones for fallback, after all, clients who implement AES-GCM will also have all the other modern crypto like ECDHE.


  Is Authenticated Encryption (AE) is the same AEAD? or they are
  different terms?


Technically yes. An AE scheme only needs to deliver authenticated encryption of the plaintext, the AEAD scheme also needs to authenticate data as well, so every AEAD scheme is also an AE scheme but not the other way around. Practically however, I am not aware of any scheme that is practically used and is AE but not AEAD (and even then, conversion would be quite simple).