TLS: Is Integrity assured when using NULL cipher
To answer your main question. 
Yes, NULL ciphersuites do provide authenticity and integrity checks but not confidentiality for the TLS traffic.

There is still a specified key exchange algorithm (e.g. RSA) for the TLS handshake protocol. The client will generate a pre-master secret and encrypt it with the server's public key for server authentication.
The session key is then derived and the MAC keys are generated from it. Authentication of messages is provided by demostrating knowledge of the MAC keys.

However, the toolkit Openssl warns that NULL ciphersuites offer no encryption at all and are a security risk. They are disabled unless explicitly included.

RFC 4785 classifies them as authentication-only ciphersuites (with no encryption).
Not sure about the your other question.