Storing DP with hydraulic levelers down

I do

Consider that your coach does not know the difference between being stored and parked in a campground for an extended period. Jacks down in both cases is fine.

Second that... They recommend it to take a load off the suspension and tires. It also makes the RV more stable in the event of high wind and such.

Just read the jacks owner's manual regarding lubrication of the jack shaft,, Usually it says something like "Wipe down with _____lubricant" (I left the blank cause if I named the wrong one that could cause problems).

Do so when you store it.

Do it again before you up jacks and hit the road.

Yes and some recommend it to take pressure off the suspension and tires.