This is a question for actual pilots.
How much trouble do I get into when I land/takeoff across the active using the grass? I have trouble with crosswinds in a taildragger like the cub.
At fields without automated weather I buzz the field to read the wind sock, then lineup downwind for landing, ignoring the paved runway.
Landing on grass at a paved airport
Well, you could have your pilot certificate suspended and/or pay a fine. The exception would be in an emergency but you would have to prove that you had no other choice.
Wind sock buzz
Wouldn't be a good thing to do from a safety standpoint. You could fly low along the runway and then return for landing but the crosswind low buzz could get you a violation. Uncontrolled airports also have regulations for pattern entry.
I better stay at dirt strips without ATC.
It is common and encouraged to overfly an uncontrolled field to look at the windsock. This is assuming you have no other means to find the active runway or wind direction. When doing so you should be at least 500 ft. over pattern altitude and announce your intentions on the appropriate frequency. Once you see the windsock you can head upwind and fly far enough away so you can safely turn back around and descend to pattern altitude and enter the downwind leg at a 45 degree angle of entry.