Short answer:
Each time it untaps, you may gain counters or mill, but not both. You may use its abilities even you've already used them this turn, as long as you can pay the cost (to tap the permanent while untapped).
"[cost]: [effect]" denotes an activated ability.
You seem to think that tapping activates the ability, but it doesn't. Activating the ability requires that you tap the permanent as a cost.
That means the first thing you do is chose to activate the ability, then you tap the permanent to satisfy the cost of doing so.
You cannot then choose to activate the second ability since you would be unable to pay its cost.
If it becomes untapped by any means, you are free to use the either of its abilities again, even if you've already used that ability. Only a Planeswalker's loyalty ability can't be used more than once per turn (and those which explicitly forbid it on the card). Any other ability can be used as many times as you can pay the cost.
"{Whenever|When|At} [condition], [effect]" denotes a triggered ability.
If these abilities were phrased "Whenever Grindclock becomes tapped, ...", then you would get both effects whenever Grindlock became tapped, no matter the reason it became tapped. You would need some external means of tapping it, though.
For example, attacking with a Fallowsage would give you to the option of drawing a card.
For example, tapping a land for mana while its enchanted by Contaminated Ground would cause its controller to lose two life (in addition to gaining mana). If that land was enchanted by two Contaminated Ground, its controller would lose a total of four life.