I'm thinking of traveling from Beijing to Moscow by rail but I want to clear up some visa issues first. With my American passport and Chinese visa I have no problem entering China or Mongolia. However, I'd like to enter Russia using my Venezuelan passport, which offers visa-free entry for 90 days. Will I be permitted to exit Mongolia with my US passport and enter Russia with my Venezuelan passport?
In general yes: it's ok to use one passport for country A and another for country B. Most border crossings consist actually of two independent check points: exit from country A (staffed and operated by country A) and entry into country B (staffed and operated by country B).
Check point A will process your exit and you should use the same passport that you have used to enter country A. Check point B will process your entry into country B and you can use a different passport for this. In many cases A really doesn't care what happens at B and vice versa.
CAVEAT: While I have done this a few times and never had problems, every combination of passport and border control point is different and may be governed by different local rules. I have no actual knowledge of your specific case and I recommend contacting the consulates for specific advice.
Yes, you can do this. However, many countries' border officers, on exit, will check your paperwork for the bordering country. If the Mongolians know US citizens need a visa for Russia, just show them your Venezuelan passport and explain that it's visa-exempt (they're unlikely to speak any English though, so knowing a little Mongolian or Russian will help)