Most of the encryption modes of Feistel Networks especially the ones used to build fixed length block ciphers just provide confidentiality .
Can we build authenticated encryption using Feistel Networks ?
The simple way to build authenticated encryption using a Feistel Network is to build a Feistel based block cipher, then use one of the many modes of operation that turn a block cipher into an authenticated encryption scheme (eg CCM,OCB,GCM).
For a good survey on the subject of modes-of-operation I would recommend this paper by Rogaway. It does not cover the very most recent designs (such as those discussed at DIAC'13) but covers the more accepted methods in a reasonable amount of detail.
Yes, you certainly can. If you want a variable-length authenticated encryption mode, then simply take any Feistel cipher in the OCB mode. If fixed-length is fine, then the following idea should work. Build a wide Feistel-based permutation (fixed-key blockcipher) $G$ and encrypt $$ C = G(P||N||K)\oplus K, $$ where $N$ is nonce, $P$ is plaintext, $C$ is ciphertext, $K$ is key.