A friend of mine has a situation similar to yours. The way he handled it was to build a raised paver patio-it's quite large and one step down (well--two steps counting the step out of/into the house) onto the patio, and then three steps from the patio onto the lawn. The level of the patio falls right between the level of the door and the lawn. He constructed it with those stackable wall systems. He has plantings planted around the perimeter and it is quite nice. You could do a deck at the midline too--would probably be a lot cheaper. I don't know if you'd be interested in this midline solution, but thought I'd throw it out there. It would bother me to walk up and down 5 steps with no landing, especially if I had to carry stuff like a tray of food or a pile of squirt guns and open and close the door at the same time. It's something to think about.

Flowering jasmine (carolina jessamine) will grow fast, look great and smell great in the spring/summer. It's evergreen down South but in zone 6 leaves will die out in winter. I'd suggest combining a couple of vines: an evergreen with something flowering so you have great interest and visual appeal in summer but won't be left barren in winter. :)

Acetone should have worked....perhaps soak a paper towel in in and leave it on the sap. Hope you learned not to trim trees as the sap is rising....they will bleed and make a mess, attract insects and sometimes weaken the tree.

A poultice is when you smear and cover a substance over the stain, and leave it there for a time, so that the poultice material draws out the stain. People often do this, sometimes successfully, with porous countertops. If you provide better pictures of the stone, people can help you with identification. If this is outdoors, then I really probably wouldn't obsess over it. Over time, you just call it "patina" and enjoy the patio.