You could make a 360 arch with the same thickness, cut it in half horizontally from the side. Select the bottom half then using the vertex tool select the outer-bottom vertices and hold and press ALT-E (I think?), which then a small window should pop up allowing you to type in some numbers in the appropriate place that should allow you to make the circle smaller from above. Find the right number you want then press enter. You can then just cut a quarter out of the arch and put it in place, should fit perfectly if the side number of the arch is not odd. Oh and you might have to adjust the vertices at the edges to match up.
I wouldn’t suggest making it out of triangles. This is the most brush consuming way to make things and to be honest it does work, very well but is rubbish for texture aligning and isn’t the most optimised way.
I would also like to mention that even though the points will be off-grid it will not be an invalid brush. Blame Valve. It’s to do with the way the polygons in the brushes are facing, they have to be natural to the brushes intended shape or it fucks up, hard to explain.
Anyway, there’s a pattern to selecting vertices in a cylinder. It’s usually something like this, it really depends on the amount of sides it has.
About your rotating door issue by the way take off both X and Y in the Properties > Flags.