"I've heard the Sunni and Shia division described as analogous to Protestant and Catholic; except that Sunni and Shia may even pray at the same mosque, because it doesn't make a day-to-day different, it just makes a difference in governance. Would you say that's a fair comparison?"
Roughly. To be honest, I've never been in a Shia masjid, and much of what I know is based on hearsay. I've heard that there are several differences between how they pray and how we Sunnis pray, but that the differences are relatively minor. Even so, there are some minor differences in how Sunnis pray (those who follow the Maliki school of thought have their arms hang at their sides while standing during salat; the rest of us cross our arms, right over left, when we stand). These differences don't invalidate one's prayer. Of course, Sunnis and Shias pray together at al-Masjid al-Haram, the Sacred Mosque located in Makkah, at al-Masjid an-Nabawi, the Mosque of the Prophet (pbuh) in Medina, and at other masjids worldwide.
"I think I understand what you're saying about proseletyzing. I am wondering how that plays into the idea we often hear about that (some) (which?) Muslims want to restore the caliphate, want a large Islamic state, want everything under sharia law, etc. It seems to me there's a difference between wanting to spread the religion, and wanting to spread religious rule. You've addressed the former; can you address the latter?"
Yes, there are some people who want to restore the Caliphate, although I think this is mostly a pipe dream at this time. (I've written about this topic on my own blog; see Bush Administration Misuses the Word "Caliphate".) Shariah is a much broader topic and would require a diary or three to explain. Some year maybe, insha'allah. ;) The thing to understand is that Shariah is a very large and comprehensive corpus of law. It covers many topics. What most Westerners worry about are the Hudud laws and their punishments for criminal behavior; however, these people haven't learned to differentiate between "Shariah" and "Hudud," the latter being a small subset of the former.
Some modern "secular" governments have incorporated parts of Shariah into their legal systems; e.g., Singapore. Here, Shariah is applied for Singaporean Muslims with regard to domestic issues (such as marriage, divorce, domestic disputes, burial, etc.). Also, Shariah with regard to Islamic finance is quickly being incorporated as Singapore embraces Islamic banking. So "Shariah" is not quite the bogeyman that many Westerners worry about.
"Related, but distinct: I've heard it said that Islam does not recognize a separation of church and state. Can you comment?"
That is essentially correct. Muslims don't really look at Islam as a religion per se; it's not just a part of our life to put away most of the time and bring out every now and then when we feel the need to be spiritual. Islam, for us, is a way of life, and many acts that we do in the course of our daily lives are a form of worship. As such, because government and politics play large parts of our lives, we don't believe that you can fully separate "church" from state. However, since the Prophet (pbuh) migrated to Medina, there has been a concern among Muslims that people of other religions need to be treated as fairly as possible and that they would not be judged necessarily under Shariah.
This is another huge topic and would require a dairy or three to cover with any justice.