If we had 84 hour days/84 hour nights, how would this effect the planets ecosystem? How much colder would nights be/hotter days would be? Can life as we know it even exist on a planet with such long cycles?
There are a lot more factors that go into life besides the length of the day. That's really the least of your worries. As to how much colder or hotter it's going to be--rotation is not even close to the only factor there.
I am going to surmise that you mean to say given that all other conditions are the same, if we changed rotation, how much would things change.
Given that there ARE actually places on this planet already which have an even longer cycle, which already do support life, it's absolutely possible--and yes, that WOULD be "life as we know it."
Answering this question is difficult because it seems you assume that this doesn't already happen on this planet, and it does. But let us assume you mean an 84 hour cycle where we would normally have a 12 hour cycle.
The thing about rotation is that it changes the range of the Goldilocks Zone--a fast rotation can make a planet too far from the sun, that would ordinarily be inhospitable, hospitable. A slower rotation--that can bring amazing changes--and a planet can be much closer to the sun and be hospitable. For instance, this article talks about how, if earth (as it is with current atmosphere and oxygen levels) were placed in Venus's rotation, where Venus is in the solar system, spinning at the same slow daily rotation, as Venus, would actually be life supporting, even though it takes 243 of our earth days for it to make a full rotation.
This is all theoretical, but I highly suggest that you read the article mentioned because it will give you a good handle on the effect it might have. Other factors can compensate for the longer spin--a thicker atmosphere for instance, could make the transition between light and dark easier as far as temperature is concerned.
Now, as "life as we know it"...it doesn't take much for evolution to go off in crazy directions. Remember that time we had a planet full of giant lizard beasts? Man. Those were the days. My point is that earth has had a wide and varied range of life on it, and there's hardly a "standard form"--and if there is, that standard form has held up to harsh conditions. We've got a lot to chose from on the adaptability scale--if you want less water and higher temperatures? We got that covered. Near constant rain and a lack of unfiltered light? We got that too...
My other point is that there isn't going to be just one cycle on this planet of yours. Our planet has a much varied day/night cycle, depending on the angle--so yours will have places on it that may roast continually or might be in darkness for much longer.