I would say that you must erode something, it is an action, so..... The senator`s support is being eroded by his unpopular positions on the major issues." (passive) The senator`s unpopular positions on the major issues is eroding his support." (active) ' but deteriorating can happen on it's own, you can't deteriorate something, it deteriorates on it's own, so... The senator`s support is deteriorating because of his unpopular positions on the major issues." 'Wearing away' will be used in the same way as eroding in this case as the meaning is very similar. Hope this is clear, the differences are quite subtle!!

Yes, any of those would work just as well

Agreed with Arron: "erode" depends on outside forces, "deteriorate" is an internal cause.