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.

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 ...

Any professional NLV (Non-Linear Video editing) app should handle this. They normally support a standard called SMPTE timecode, which is something like 3 or 4 digits of precision per second. Software in this category would include Sony Vegas, Final Cut Pro, Adobe Premiere, etc.

For Linux, try Cinelerra (fairly outdated and horrible to install), KDEnlive, or Pitivi. I'm sure that Cinelerra will do it, should you be able to get it running, but not so sure about the others.

This isn't free ($19), but Womble MPEG-VCR has frame accurate times with frame by frame stepping.

If you do a search use the key words "frame accurate", I'm sure there must be something freeware, especially for Linux.

Hope this helps.