When can you counter a suspend card (when it is suspended or when the final time counters get removed) and would I need to play a counterspell (counter a spell) or a stifle (counters an ability) to do so?
Each of your subsequent upkeeps, a triggered ability removes a time counter. This can be stopped by a card like Stifle. (Note: Stifling the "remove a counter" trigger won't permanently stop the spell, you will simply delay the spell by a turn.)
When the final time counter is removed, another triggered ability (which can also be the target of Stifle and friends) will cast the suspended spell upon resolution. At that point, the spell will finally be on the stack, and a legal target for cards such as Counterspell.
Suspended cards have three abilities, two of which may be countered. To understand why, we first need to understand how to counter a spell, and the various abilities associated with the
701.5a To counter a spell or ability means to cancel it, removing it from the stack.
Now we know that a spell or ability must be on the stack in order to be countered. Now we explore the three Suspend abilities, and determine which can be countered and which can't.
Suspend N—[cost]” means “If you could begin to cast this card by putting it onto the stack from your hand, you may pay [cost] and exile it with N time counters on it. This action does not use the stack
The first ability is a special action that does not use the stack. It cannot be countered or even responded to at all.
At the beginning of your upkeep, if this card is suspended, remove a time counter from it
The second ability is a triggered ability that uses the stack. It can be countered by Stifle. This typically only serves to delay the suspension by a single turn.
When the last time counter is removed from this card, if it’s exiled, play it without paying its mana cost if able. If you can’t, it remains exiled. If you cast a creature spell this way, it gains haste until you lose control of the spell or the permanent it becomes.
This final ability is also a triggered ability which puts a spell on the stack. The triggered ability may be countered with Stifle, or the spell may be countered by Counterspell. If the ability is countered, the previously suspended card remains in exile, where it is typically never able to trigger again. Countered spells go to the graveyard.