If I reinforce an ally, do my soldiers die or get hospitalized?

Your soldiers should usually be automatically sent to your hospitals if they are defeated. However, with these types of games, there are some buffs on more advanced players that enables them to automatically kill a percentage of the soldiers that they defeat -- regardless of the availability of room in your hospitals. Be sure to check on the battle results after your troops have been defeated -- to gain a better understanding of how they lost, how you lost power, and to help you plan on how to improve your situation.