I wouldn't bother with that machine, they do not have enough pressure to spray any thick materials.
Get yourself a good sheepskin long pile roller, hog enough paint on and sooner or later it will fill all the holes up. :)
Sounds like the walls were painted with a vinyl type of paint which is a big no no for new plaster, all you can do is scrape off the chunky lumps, block fill the bits where the chunky lumps came from and repaint.
Or just moan to whoever you got the flat off and tell them to do the snags.
I would 99.9999999% believe what it says on the tub of emulsion rather than a receipt.
How big is the tub size for a tenner? 10lt, 5lt, 4lt?
I wouldn't use vinyl emulsions on fresh new plaster or drylined walls/ceilings, use what you have.
Yes, go ahead and paint your Crown over the...