However, in the interest of laziness, I'd rather not have to go through explorer opening, scrolling up to the desktop and clicking, if I could have just clicked the save icon and be done with it
Also, I wouldn't mind being able to print directly from the capture instead of having to save it first, open with a graphics app and then print
I don't think that can be done. It is saved to the clipboard by default, so after taking the snip you can open paint, paste, then print. Not what you are trying to do, but is a work around.