How does the GCM encryption limit affect TLS connections?

According to the Wikipedia article on Galois/Counter Mode, GCM is limited to encrypting $2^{39} − 256$ bits of plain text (64 GiB) for any given key and initialization vector. In light of this limit, what considerations should be made for a TLS 1.2 connection with an AES GCM cipher suite?

1. Should the session/connection be renegotiated or manually closed when the limit is reached? Or does the TLS protocol specify that the connection will be automatically closed or generate a new IV when the limit is reached? Or is it implementation dependent?
2. Does the limit apply separately to each peer (server and client) or does the limit apply to the total of all data encrypted on both sides of the connection?
3. Will resumed sessions use a new key/IV combination thus resetting the count against the limit?
4. Do AES key size or other cipher suite options affect the encryption limit or answers to the above questions (as long as it's AES GCM)?

Note: Using TLS 1.2 with AES-256 GCM encryption and ECDHE-ECDSA key exchange/agreement via SslStream for .NET 4.6.2.