Hi John,
Photo Mechanic does not place photos into any sort of database. It views folders of images where they are on your computer. When you are looking at a Contact Sheet in Photo Mechanic (The window with thumbnails), you're looking at a folder on your computer. Most basically, to remove a photo from what you see, you would remove it from that folder. When Photo Mechanic uses the term "Ingest" it generally means copying from a memory card to a folder on your computer. You can think of "Ingest" as a fancy word for "copy" (most often from memory card.) It is possible to "Ingest" from folders already on your computer, but what would be happening in that case is that they are being copied from that folder to wherever you have told Photo Mechanic to browse.
For your situation, there are a number of ways to work that could be helpful.
A) You could either select & move the "keepers" to their own folder and view them there, or you could select and move the non keepers into an "Alternates" folder.
B) You can rate all your photos with color classes or stars and use the filters at the bottom of the Contact Sheet to only show highly rated images.
Ctrl-y (The keyboard shortcut for copy/move) is your friend here. And there is an option in that dialog to automatically create a subfolder in whatever folder you're currently looking at.
There are other options as well, but those are common places to start
Now for your Desktop problem, if you are looking at your desktop folder in Photo Mechanic, you will see all the files there. They are not ingested into Photo Mechanic, but simply shown when you look at the Desktop. If you have images that are on your Desktop, you can choose to sort by filetype to see them together. I would personally recommend selecting the images and moving them to your Pictures folder which you can do with drag and drop.
In your Photo Mechanic Preferences (ctrl-comma) under General, you can specify what Photo Mechanic does on startup. Right now it sounds as if it is set to "Open last contact sheet" which would be the Desktop if that is the last place you viewed. One thing I would also suggest is going to the Navigator, navigating to your Pictures folder, then dragging the Pictures folder from the Navigator into Favorites so that you can get to it quickly no matter how you start PM.
I hope this gives you a good place to start as you figure out the workflow that works for you. (And I hope the way I have tried to explain it has not made things worse :-)
Let me know if you have any follow-ups and we'll do our best to clear things up.
-Mick