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In the pics, it looks like the height started out okay then it went lower? Maybe thats just camera angle, if not then either the bar was not level to the table, or the bit slipped in the chuck. Are you using a end mill holder with side set screw? Holding in a drill chuck is a bad idea, the bit will move on you. And I would make a number of light passes, moving in or down a bit at a time depending on which is your 'fixed' axis.
At that size one pass should not be a problem.
Re. which to make first, the mouthpiece or the rod, I think I'd probably do the shaped holes in the m'piece and gland first, get them as accurate in size and position as possible, and then aim to do the fitting on the rod. The holes need to be nicely aligned in concentricity with the bore of the cylinder, so if that has been got right by careful machining, it might be better not to wander away from that attainment by doing much in the way of subsequent filing of the holes. The rod will be easier to take down to a sliding fit and a nice finish by light draw filing along its length.
A bigger radius cutter, and you can make your own round bar! Always fun to learn a new tool...