I frequently use slatted racks, and I really like them for helping the bees control the hive temperature, but as for preventing swarming, not in my experience. I guess it's possible that a hive would go a little bit longer before deciding to swarm, since the slatted rack provides a little bit of extra space, but hives that are planning on swarming will swarm, and a little bit of extra space and wood isn't going to stop them.
Keeping only 3 hives is certainly possible; we usually recommend newbees start with a minimum of two, so three is definitely okay. Keep in mind though that having any exact number of hives at all times can be tricky. For example I average around 5 hives, and I want no more than 10. So last summer my apiary had 8 hives in it and right now I'm down to 4. Big colonies will want to swarm and you'll need to split them, and weak colonies will die or be combined with strong ones, so the amount of colonies you have at any given time will usually vary.