Hi,
To stop it searching your whole system, you don't use the option to get Photos and Videos by searching (see under File menu). When you first start using elements, I think it asks you then if you want to search your disks. So normally, the searching is not invoked. What can be a problem is the watch folder option. If it is switched on, make sure that the folder to be watched is not at too high a level - probably ok if set to C:\Users\Me\Pictures\Adobe as most of your images are stored under that main folder.
There is no "save" or "Save as" in the organizer because you don't save the images you just index them. If you download an image from Facebook you have to download it to somewhere. If you are using a browser to download the image it will probably go to your download folder unless you specify somewhere else. If it appears in your catalog, it has probably been detected by the watch folders process and added. The image has already been saved to your hard disk sao you don't need to do a save. If you want the image to be in a particular folder then you can use the organizer to move it.
There is a facility in the organizer to email directly from the application. If you see a thumbnail that you want to send, you select it and then share by email attachments of as a Photomail. This tutorial will explain a cit more.
Emailing photos from Photoshop Elements 12 | Learn Photoshop Elements 12 | Adobe TV
Brian