Bicycle Mechanics - Confused about freewheels

I have a bike with a very rusty and worn freewheel. How do I tell what replacement type to get? This is an old made in Taiwan bike with a 6 cog rear (12-speed bike). The rear deraileur is Shimano Tourney. Also the round plastic thingy which goes alongside the large cog is chewed up, is that a standard replacement part. I noticed some bikes have a metal ring instead of plastic.

Does the fact that this is a Shimao derailleur mean the freewheel is Shimano? Could I replace a 6 cog freewheel with a 5 or 7 cog one? Is the width of the hub a factor? Short of going into a bike shop, is there any way to figure out what I need?

There are two types of 6-speed freewheels: standard, which require a 126mm (over locknuts dimension, O.L.D.) rear axle, and "ultra," which can be used with a 120mm rear axle. To figure out which you have, simply remove your back wheel and measure the distance between the inside faces of the rear dropouts. If you have a 120mm rear axle and friction shifter, you may substitute a 5-speed freewheel. If you have a 126 and friction shifter, you may substitute a 7-speed, particularly if you toss that damaged plastic spoke protector. (You can probably find a replacement spoke protector if you really want one; I never use them.)

Since your derailer is Shimano, I'd definately suggest you get a Shimano freewheel. I'd be surprised if Suntour or any other make would shift as well as a Shimano one (unless you have friction shifters (no clicks), in which case they'll all probably work).