Yes, cold weather will definately have an effect on the tarantula.
The spider would probably become less active and therefore eat a lot less and increase its time between moults.
Is this a bad thing though? IMO, no. In the wild there would be cold spells and this I would think serves as a bit of a "recovery" period for the spider. I generally keep my spiders at room temperature, which means in the winter my spider room would drop to around 15 degrees I estimate. The spiders usually cut down on their feeding (Saving me from buying crickets/roaches) but also just giving them a break from their normal summer routine of eating and moulting. To add to this, when it comes to breeding my spiders, the cold spell comes in useful for the process that I use to breed them. Giving them seasons as they grow up to maturity also helps me in the breeding process I think. As long as you are comfortable with the temperature in the room without dressing like an eskimo, the spider will be fine.
As long as the temperature doesn't drop below 10 degrees for extended periods, the spider will be fine and eat less.
As for leaving crickets in, I would not recommend it (Especially if you think the spider is in premoult). If you need to, leave the cricket over night, but remove it the next morning if the spider hasn't eaten.