These are lawful traits, yes, but you have mostly left out the "good" portion of a paladin's alignment. Valuing freedom is a chaotic trait, yes, but it is also a good trait. Good behavior that is coerced is not truly good behavior. In order to be truly good and to be rewarded as good it must be freely chosen, which requires that there also be an option to choose evil.
Respect for the dignity of sentients, part of the description of "good" in the 3.5 PHB, also requires allowing them to make their own choices.
A paladin can argue that his lawful aspect requires him to choose the good of the many over the good of the one, but his good aspect also requires that he sometimes choose the good of the one over that of the many. (Hat tip to Star Trek III)
No, no, and no. A lawful good deity understands that devotion cannot be compelled. Therefore a lawful good deity will not mandate worship, and certainly won't allow his or her followers to hand out draconian punishments for those who won't be coerced. You are describing a lawful neutral or lawful evil system that wants to call itself lawful good. Any paladin working for such a system would fall.
A good DM could still put together the scenario you describe, with the empire claiming that they do have real paladins, but there would be no true paladins on the side of this empire. The clerics would be deriving their spells from some other god, possibly even while being deluded that they were worshiping the lawful good deity.
Like all good science, it would require strict controls and repetition to provide accurate results, but it could be done. Repeat an evil action often enough and you will shift your alignment to evil, even if it's petty theft or just lying, and that alignment shift will be detectible with detect evil.