My wife has a Schwinn Windwood. The sprocket on the rear wheel keeps popping out. How can I correct this?

You have conflicting info. If there are multiple gears then there is a derailer. You should take to a bike shop where they can look at it. It is probably a very small easy adjustment.

You have to tighten hub axle and leave very little play there.

It's loose on the hub. Special tools are required to tighten them. Bike Shops have them and will possibly perform the repair for less than the cost of the tools.

http://www.parktool.com/blog/repair-help/categories/cassette-and-freewheel-service

Then check the shifting...

http://www.parktool.com/blog/repair-help/categories/derailleur-systems

Word to the wise: don't buy bicycles from anywhere but a professional bicycle shop. Box stores don't usually have the skills for proper assembly, setup or follow-up.

Take it back to your dealer unless you have the correct bottom bracket wrench.

Why not call a Schwinn dealer and ask if they can order it for you? You do realize that you void any warranty by altering you bike, don't you? But, if you don't care, look on the www.schwinn.com website and find a Schwinn dealer near you.

The pedal is threaded opposite the right side pedal. Try screwing it in the other way.

To insert your right pedal you had to screw the pedal toward the front of the bike, the same will be true on the left side. Before you start however make sure your left pedal has an "L" on it indicating it is for the left side.

If the chain has just slipped of the best way get it back on is to physically lift it into place at the top of the chain ring (the cog where the pedals are).
Then turn the pedals in the direction you would usually pedal (you may have to lift the back wheel off the ground).
This ought to pull the chain back on and let you get back on the open road.

You may have knocked the gear derailleur when fixing and re-fitting the rear wheel. This can knock the derailleur out of whack slightly and will need proper adjusting by your local bike shop.

Does the rear wheel go on such that you can adjust it to take any play out of the chain? If so, turn the bike upside down or put it on a bike stand. Loosen the back wheel and pull it so that the chain is tighter and then tighten the wheel. There are other possible causes but I suspect this since you recently removed the wheel. Also I'm not sure how often the chain falls off. If this doesn't work, ask again and let us know how often its coming off. Good luck!

You have it on backwards.  I had the exact same problem.  Turn the fender around so that the arms that connect to the axel are pointing forward.  They are going to be almost parallel to the ground.  Easy Peasy lemon Squeezy.