Hi, kmail. Welcome to FlyerTalk!

Newark would be a more reliable option, since you have a longer layover time in which to make it through Immigration & Customs. Two flights (as opposed to three) reduces your chance of missing a connecting flight.

Yes, bags are always checked to your final destination if all flights are on a United itinerary. Even if they're not (let's say you have two different itineraries to get to a destination) you can request to have the bags interlined by by showing the agent your itineraries when you check in. However, since everything is on United for this, your bags will be checked all the way through.

Please note: As with all international flights that transit through the United States (except a few cities where U.S. Customs is outside the U.S., like Dublin or Vancouver), you'll need to collect your bags at the first stop in the U.S. (Houston or Newark), carry them with you through Customs, then check them again for the remaining flight(s). This process is usually straightforward, as the Bag Drop area is right after Customs in most U.S. airports.