Elliptic Curve is DH function or PKI? can we reuse same ECC key on TLS for long terms or it must be used just once? (i mean can we use ECC like RSA?) 

is there patent free ECC implementation ?
Yes, TLS supports ECDSA for signing and ECDH for key exchange. You can read about it RFC 4492. Notice you don't get RSA, but you get signing and key exchange which are major functions of TLS.

Related material:
How do I create an ECDSA certificate with the OpenSSL command-line
SSH key-type, rsa, dsa, ecdsa, are there easy answers for which to choose when?

As far as patents go, see this question. I am not aware of any ecc libraries that claim to not violate patents. Bouncy castle says it is up to you to figure that out. Openssl's ecc implementation was supposedly built to avoid patent issues.

Is elliptic curve a DH function or PKI
Neither really. There are elliptic curve variants of ElGamal and RSA. Depends exactly what your definition of PKI is though as you can have ECDSA certificates with a public and private key. You can then start a PKI by signing other certificates, etc.