Go frame-by-frame through a movie with a precise timer

Give Avidemux a try. It should be able to do what you want and is freely available for Windows and Linux.

I don't know if this feature was available at the time the question was asked, but the open source MPC-HC (Media Player Classic Home Cinema) video player definitely has both these features available now.

There is a context menu option to enable millisecond precision for time indicator, and you can easily step between frames in both forward (CTRL+Right) and backward (CTRL+Left) directions.

You can also jump to exact time point OR frame in video file via Navigate -> Go To... menu (or CTRL+G).

Well, seconds are integers. Video players rarely go below them to more precise measurements.

But if you know at what frame you're currently at (which is fairly commonly displayed), and you know the frame rate, you can usually calculate it to much more of a second. Or you can skip frame by frame while on a time measurement, and the moment the next second "hits", switch to frame measurement and then calculate the percent of the next second in which the particular frame occurs.

... just some crazy ideas ... have never had the need for this, so all this is untested ...