Draw claims can be made at any time as long as the conditions are met. So, if a position appears again for the 4th time, for example, the draw can be claimed but if it isn't claimed can't be claimed again until the position is repeated again.
Same goes for the 50 moves with no captures or pawn moves. It it has been 67 moves and suddenly one side has a mating attack, the draw can still be claimed if the no captures or pawn moves have happened in the last 50 moves.
It is clear that when offering a draw, it is up to the opposing player to either accept or reject the offer. On this site, this is done by selecting a check, an X, or any other appropriate symbol.
However, when there is a condition for a draw (such as fifty moves without progress or position repeated three times), from my experience, hitting the draw button automatically generates a draw.
I wonder, though, for how long would this automatic draw (without acceptance from the opposing player be required) be in effect on this site? Is it only immediately after a position has been repeated three times or fifty consecutive moves have been made without progress? In other words, if forty moves later a player has taken a decided advantage, would the losing player hitting the draw button automatically generate a draw because of the earlier condition?