I'm kinda torn here, Walt. I like helping people out with learning music - that's why I teach for a living. I've got no problem with the first part, different strategies for modulation.
The smoothest way in this instance is through the tonic minor: go from C to Cm. Cm is the relative minor of Eb.
Other strategies would include these:
1. Through secondary dominants. In C, you've got G7 as a dominant chord, which leads to C. If you substitute C7 for C, that has a tension that leads to F. You can keep going, and get from C to Eb by this route: C7 -> F7 -> Bb7 -> Eb
2. Through common chords. There aren't any chords in both C and Eb, but you can do it in a couple of steps. The principle chords of C are C, Dm, Em, F, G (or G7) and Am. In Eb you have Eb, Fm, Gm, Ab, Bb (or Bb7) and Cm. If you took something like the above strategy to go from the key of C to the key of F for a bit... well, F has a Bb chord. You can use that to pivot directly into Eb.
3. Modulation through a sequence. You could do C-Dm-Em... D-Em-F#m... Eb-Fm-Gm. You can include a Db sequence if you want to do the whole thing chromatically.
4. You can just jump to it. Nothing says key changes must be smooth, and at times a jarring change is best.
The second part of your question I have a problem with, because it doesn't strike me like you're asking for help understanding modulations... it seems like you're asking for help with your homework. If you're the one writing the song, YOU decide how many beats per chord. Whether or not it modulates has nothing to do with it.
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