1) yes, the repetition is okay
2) it does tend to emphasize it, since it (literally the it) is repeated three times. It seems mostly a matter of style. If the noun (rather than the pronoun) was repeated three times (test my theory, learn from my theory and improve my theory), then that would really be repetitious and not win many awards for eloquence.
3) yes, the order is okay as you have it
4) again, it's a matter of style. A writer might order the verbs just by how they sound, or perhaps put the least important, or the one he wants to stress the least, in the middle. In that sense, it gets the least attention, just like a middle child. ;) On the other hand, putting learn from in the middle prevents two one-word verbs from following each other, thereby putting in some variety of cadence.
Finally, you might want to consider improve upon rather than improve. But that partly depends on what the noun is. If it were theory, I might well use improve upon. One thing improve upon does is build up the syllables of the verbs by one each time. You go from one to two to three syllables. Which is why choice doesn't just depend on meaning, but on sound, cadence, rhythm, etc.
Edit: improve upon has four syllables, obviously. I miscounted. Or, actually, I had in my head that improve is one syllable, since the first syllable gets deemphasized in pronunciation.
Lastly, using your three verbs without repeating it can sound a bit strange, but once again it is a matter of style, not whether it is grammatical, which it is.