Frequently in sketchup I do things that should result in coplanar shapes but end up with shapes that are slightly off and aren’t on the same plane. Is this a bug? What is the proper way/workaround to prevent this from happening?
These shapes are coplanar. The very unique way in which you have created the two circles by starting with a line that was moved and then bisects the first circle invokes a version of the so called “bow-tie” problem. Not really a bug but a situation in which SketchUp is confused about which surfaces to merge. If you select all and then right click and choose intersect you will see that they merge back together. It’s a rare occurrence, if you make the first circle without moving the line, it will not happen. Can I ask why you are making and then moving the edge in the first step? This seems unnecessary, simply start with a circle, type in the dimensions you want?
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Intersect doesn’t fix it. The way I did it was a habit from before I learnt you could input formulas into the dimension box (edit, wait, I thought you could, but entering 16.8/2 just results in a circle with a radius of 16.8)
I just tried it now, first by laying down the line as you did at the start, which resulted in the coplanar problem.
I tried again by just making two circles without the extra geometry, and I was able to delete the inner circle from the outer one.
The only way around this is to stop using the line tool.
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You must delete the bisecting line before intersecting the geometry.
No, the circle tool does not accept diameter, only radius.
If math is not an option and you must enter diameter by drawing a line the way you have, then cut it in half before your circle or move it up out of the way and reference it so it does not bisect the circle.
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You can use maths to enter the radius as a diameter divided by 2 but the diameter must be a whole number, 20/2 gives a radius of 10 units but using 20.2/2 will be seen as a radius of 20.2 units. The same goes for the line tool which will divide a whole number into however many pieces you want - 20/3 returns the correct distance. It will be displayed with as many decimal places e.g. 6.66667, of the accuracy you set in the model info section.
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If it’s easier for you to work with diameters instead of trying to work out a radius (which isn’t always intuitive or convenient), one workaround would be to make a square in the diameter you want, make it a group, then inference its center with the circle tool and snap to the edges. Then, delete the square and carry on.
I do this whenever I have to work with fractions of an inch because working out the radius of a 21/64th hole is a pain.
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FWIW the radius of a 21/64 in. hole is 21/128 in. 
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If only SketchUp would display that instead of “~ 3/16"”
