Yeah, if you look under run it says you lose your dex bonus. There is not an actual double move action, you can simply exchange your standard action for another move action. It is not noted to incur any penalty to do so, so you do not.
Yup. Specific trumphs general. But if there is no specific rule to change general then general stays in place. The general rule is that you get dex to AC. Jogging (double moving) has no specific rule that changes general so general stays.