1) Yes, but Chaplains are strictly religious, not legal, entities. Further, they are employed by the federal government and would have no ability to marry even if there were some legal status. Marriage is a function of state governments.
2) It's important to keep in mind that military is a case where the free market is generally not at work. Servicemembers take long contracts that they can not negotiate their way out of and surrender all control of most of their lives. In exchange, the military generally takes responsibility for a standard of living on top of their salary. In Spc Lloyd Baker's case, he is probably assigned a barracks at his home base and hence not eligible for a housing allowance. Once he is married, the military recognizes his need to cohabitate with his wife and now pays him a Basic Allowance for Housing. Keep in mind, as a military spouse his wife will be unable to pursue most professional careers due to having to move with her husband at the beck and call of the military.