I get that harmony is the vertical dimension of music (thinking in terms of sheet music or a piano roll view in a DAW) and melody is the horizontal dimension.
If you're really thinking about sheet music in standard notation, or a DAW piano roll, then the vertical dimension is note selection (which in itself is an abstraction of pitch), and the horizontal dimension deals with rhythm.
In this model, melody can't be thought of as a single dimension, because it needs to be defined both in terms of pitch and rhythm. That's also the case with a harmonic progression.
The instantaneous harmony of a piece - "what chord is playing now?" - could be thought of as being defined by a set of values on the vertical dimension.
It's worth noting that this just a way to view music - it isn't 'the correct' way. If you look at a piano keyboard, then suddenly note selection is the horizontal dimension. And there are other important aspects of music, such as timbre, that could be considered 'dimensions'.
how is it still possible for the rhythm to vary while holding the melody constant?
It isn't. If the rhythm of a melody is changing, then the melody is changing.
To offer another guess at how melody may not completely define the horizontal dimension, is it the case that melody is only concerned with the order of notes, while rhythm takes credit for their duration? For example, if I specify a string of notes such as A, C, E, G, F, C#, A, B, A, C, A, does this string, in and of itself, constitute a melody, even though the string might sound very different depending on the duration of each note?
It's not a crazy idea to think of the order of notes being one aspect of a musical part, and their rhythm being another aspect. In fact, that's pretty close to the dimensions we see on sheet music. However, your definition here doesn't agree with what people mean by 'melody', which refers to something defined by both note selection and rhythm.
How do rhythm and melody compete and cooperate to define the horizontal space of music?
It makes no sense to talk about rhythm and melody competing, because they're different levels of concept: melody is a higher-level concept defined partly in terms of rhythm.
My grade school understanding was that rhythm was the percussion's contribution to the horizontal dimension and melody was the non-percussion's contribution to it, so that a melody would not completely define the horizontal dimension, but only the role of certain instruments in occupying it, but I don't think this explanation is correct, because people attempt to illustrate the concept of rhythm on non-percussive instruments as well.
I think you're correct that that's not correct - again, it seems to be mixing up lower- and higher-level concepts. We also need to be careful because some percussion instruments can play melodies!
Perhaps one model we could use is that unpitched percussion, like a drum kit, navigates the dimensions of timbre and rhythm, while pitched instruments (that can play melodies) can traverse the dimensions of note selection, rhythm, and timbre.
But again, that would just be 'a' model - not 'the correct' model. You always have to be careful about statements saying that 'these are the X dimensions of music', or 'there are X aspects of music', because various models are possible. And as the saying goes, all models are wrong (but some are useful).