[Boromir88]

1. Was Gothmog (orc) known to have any connection at all to the other Gothmog (balrog) ?

First we don't exactly know what Gothmog was, for we only receive one line from him in the books. That he was lieutenant and 2nd in command. What Jackson shows, I have no idea where he came up with Gothmog the limping, warped, bubbly, hairy...sloth thing. I doubt he was an Orc because he was commanding Haradrim and Easterling forces. Someone that high up in Sauron's ranks I doubt he would trust an Orc to lead, and seeing as he was leading Men, I think it's most likely. But since we are not told the possibility that he was an orc is still out there.

The only connection between the two Gothmogs is their names, they would not have known eachother or anything. Gothmog was Melkor's second-in-command, Lord of the Balrogs and all that hoopla. And Sauron seems to mimic that and name his second-in-command Gothmog.

2. Are there possibly more than 1 balrog that existed in the third age besides the one that the Fellowship encountered in Moria?

No, Durin's Bane was the last Balrog left. In earliest drafts Tolkien talked about armies of Balrogs and them being a race of their own, this was when he concieved balrogs as being much weaker.

However, later on he changed his mind as he made Balrogs stronger (saying they were now Maia) and he limitted the number of them:

"There should not be supposed more than say 3 or at most 7 ever existed"~Home X: Morgoth's Ring

And we are also told that Durin's Bane was the last of them:

But it is here found (there is usally a hang-over especially of evil from one age to another) that one had escaped and taken refuge under the mountains of Hithaeglin (the Misty Mountains).~Letter 144

During the War of the Wrath the remainder of the Balrogs died, but one (Durin's Bane) escaped and took refuge in Moria.