If I am a Lv.60+3 Demon Hunter playing through the campaign on Inferno and want to go back and play through the campaign on a lower difficulty, will I, for example, lose any stats on my character?
No, there's no penalty for changing the difficulty. You can go back to Act 1 Normal and your character will be exactly the same.
The downside is that the rewards are based on where you are in terms of acts and difficulty, so you will gain much less experience and find really poor items at lower difficulties/acts.
Switching quests may cause you to lose progress in your current quest regardless of whether or not you change the difficulty setting.