Oh yeah.  If you're not careful,  Hobby-grade CA glues in thin and medium viscosities, such as Thin Zap and Zap-A-Gap, will wick all the way from one side of the wood and dribble out the other, especially if you hold the pieces so the joint is vertical and you allow gravity to pull the glue down through the joint.

In fact, the only way to keep that from happening is to hold the piece so the joint remains somewhat horizontal. Otherwise, the glue will simply flow out the other side and downward along the path of least resistence, just like water. 

For very tight-fitting joints, thin CA works best. For looser joints, medium CA does the trick.

The only real problem with wicking CA into a wide glue joint is assuring even coverage.

For your bridge, I would not use kicker to glue the bridge in the manner I described. I was just speaking generally about gluing two pieces of wood together.

Here's how I would attack gluing the bridge back down. Buy thin and medium CA. Also buy some sort of tips that will allow you to reach deeper into the glue joint than, just the bottle itself. If the glue bottles already have long tips, then you won't need to buy any. (See: http://www2.gpmd.com/image/p/paar3000.jpg)

Lay the guitar flat on its back. Use something shim it to keep it from wanting to roll around.

Spread the bridge apart from the top and use sandpaper on the finish to give the glue some tooth.

Squeeze a bead of glue around the periphery, about a 1/4" in from the outside edge. Squeeze another wavy bead that connects the outer bead to itself. The idea is that you want the glue to spread itself out and cover as much area as possible when you push the bridge joint back together. You may be applying glue on top of the finish and to the top wood.

Push the bridge joint back together and clamp it. Wipe off any excess glue that squeezed out of the cracks. Let this sit for 24 hours so the glue can dry. If you want to speed up the drying process you can spray on kicker after the joint is assembled. This will cure the edges of the glue. As the glue cures it will start the catalyzation process on the inner parts of the glue.

Above all, make sure that all the wood joints are back in their original orientation to each other before the glue starts to cure because once the curing process begins, there's no going back without breaking wood.

1) Blowing out dust is sufficient. Unlike most other glues CA doesn't require clean, dry surfaces. I've used it on dirty, oily model airplanes that needed field repairs and it worked fine.

2) If you are able to get tips like the ones I pointed to above, you should be able to get the glue into most of the joint.

At this point, your best bet is to get some glue and some scrap wood to experiment with. That will help you decide the best way to glue the bridge back down. It might take applying medium glue under the bridge, squeezing it together, then applying thin CA along the edge to wick into where you couldn't reach with the medium.

Thin and medium both have the same strength when cured. The only difference between the two is their viscosity and their ability to wick and fill gaps.