2 colors have a rectangle around them. 1 is transparent black and the other is white
These indicate the background and foreground colors, ie. the colors assigned to the left and right mouse buttons for painting. This piece of information is not saved in the image, it only exists while in the editor.
1. How can I switch to 16 color mode and if I do, out of the 256 colors, which turn out to be the 16 colors that get saved to the file?
Grafx2 itself doesn't support less than 256 colors, so you can't "force" a limit like 16 colors. When you save as GIF or PNG, it always saves the entire 256-color palette that you see in the palette editor. In any case, if your image uses only colors number 1, 7, 23 and 135 for example, it "is" a 4-color image. It's only for your own convenience that you usually arrange the palette so the used colors are at the beginning (0, 1, 2 etc) and in a useful order (for example dark to light)
transparent color?
The transparent color is one of your palette: Your image is in fact either 256-color or 255-color+transparency. If you want part of your image to be transparent, click the "layer" button, activate "Background transparency", and select which color number acts as transparent. While in grafx2 you will not see the difference, but in the saved GIF or PNG, the marked color will act as transparent.
All other colors are black now and there is a thin white line next to color 0.
In the Palette screen ? It indicates which colors are actually used in your image : Click the "Used" button to toggle these indicators on and off. Colors are marked if there is at least one pixel that uses them. It's very usual to have duplicates in the Palette (for example many colors "black"). These indicators can help you check if you have accidentally used several of them.