1. By necessity, it must have a CL starting at level 1, unlike the regular Ranger. Whether or not it then progresses CL=level or half level is largely up to the DM. Personally I think it should progress per level, but the strict RAW is likely that it progresses normally (that is, at CL=1/2 level).
2. Sword of the Arcane Order allows you to prepare spells from a different list in the spell slots granted by your class, so you still use the Mystic Ranger's casting mechanic. This includes no spell failure.
3. See above. They should be considered divine spells, as the feat doesn't state otherwise. Compare Rainbow Servant's capstone, which specifically states that Cleric spells not regularly on your list are cast as divine spells.