Well, the answer to your first question is yes. I think the first sentence is fine, but the second sentence might be ok too (though I'm not sure you can use "mou" in that context). It just depends on what you want to say. If you're trying to put something in the past tense, then yes, you change the verb before "hazu" to plain past form. But if you leave it in the dictionary form, it just makes it future tense.
I think this should clear up any misunderstandings:
http://www.guidetojapanese.org/should.html