At some points, the sound randomly stops working, and doesn't start working until you either leave the game or finish the current one. Is there any fixes available for it, or does it need to be fixed by a patch?

Try disabling all VOIP in the game.
Also browse to: Steam\steamapps\common\brink\directx, Run DXSETUP.EXE and let it finish.
You may also want to check for updated drivers for your sound card.

If this is happening on the Refuel map, then that's a known issue that will require a patch.

Heard a rumor that turning of ALL Voip options prevents this bug from happening, My voip options have been of this start, and I have yet to encounter the bug.

I can collaborate that turning off VoIP options seems to prevent this issue. I'de encountered it previously and after turning off all in game VoIP settings I haven't had the issue since. Although there are plenty of other issues to ba had