The problem now is that they have been bloating the hard drive on my laptop. The laptop hard drive has a capacity of 186 GB with only 1.13 GB now Available.
You need to have at least 10% free space left to work with Adobe PS or any application in common. Most applications use space to write temporarily files to (as they do in RAM which you should have at least 1 GB RAM but preferable 4 GB RAM to use PS and Bridge in a usable speed.
The 10 % rule is an indication, having a 1 TB disk it should do, but having only 186 GB HD at least 40 GB free space would be better.
How do I use Bridge without running into storage capacity issues?
Go to the folder panel and from this panel select your External HD and find the folder you want to work in. Bridge reads the files and starts building cache. You are able to work on those files and open them from this disk in PS and also can save the files again to this external Disk as well as renaming and adding metadata etc. in Bridge without problems. Only restriction is you have read and write permissions for that disk. And performance might be a bit slower instead of working from your internal disk but with that less amount of free space your external disk probably works faster.
Where does the PICTURES folder store these files?
Don't know what you mean with that. If you copy your files from the external disk to your internal disk the files should be in the folder you copied them to. But maybe you mean the cache for the files you view in Bridge?
Every folder you point Bridge to and having content lets Bridge starts building cache for those files. If you have set the previews to High Quality (default) it will build additional cache data for those previews and store it in a cache file so you don't need to do this every time again. This Cache file is stored on your internal disk by default. (user library/ Caches/ Adobe/ Bridge CS5 / and inhere is the folder Cache.
Or check the Bridge Preferences Cache section for the exact Location. In this preferences you can also set a new location (to your external disk, but again this might be slower in performance) to save space on your internal HD.
Is there a way to point Adobe Bridge to a storage folder on my external hard drive?
See above with quote 2
How should I manage a growing list of images without running into capacity issues?
To be honest, having a Mac Book Pro with only 186 GB is not the best way to go with todays standards. Both Hard Disks and RAM are relatively cheap these days. I have an old Mac Book and replaced the 160 GB HD and 1 GB RAM myself with the help of a very easy instruction video provided by OWC computers. I also bought my HD (now 500 GB) and RAM (4 GB) at least 1 year ago (also via OWC) and only had 75 dollar to pay. Just check it out your self using Google and max everything out if possible, you sure will benefit from it