ProMar200 Zero is decent paint. I painted my house with the Eggshell (supposed to be Satin?) sheen over Level 5 smooth walls and love the look. This was a remodeled house with all new gypsum board. I have little kids that literally bounce off the walls and the paint is holding up well and touches up very well. The walls have seen dirty little fingers, pencil marks, a bit of blood from nosebleeds, impacts from flying toys,... you name it. It is certainly not top-of-the-line, of course. Cashmere is better quality than ProMar. I would personally do low lustre or pearl everywhere, but that's only assuming your drywaller leaves a perfect substrate. I'm not a fan of "flat" walls. Some SW products have less sheen than stated, but Cashmere is actually higher in sheen. Paint a 2'x2' sample on your wall before committing to a sheen. Good luck.

My DD used Benjamin Moores Stonington Gray but had it mixed in Ace paint. So you can use other manufacturer's color mixed in Sherwin Williams paint. Here's how it looks in a well lit room with dark blue accents. It looks a little darker in my DD's hallway which is a low light area. I recommended the Stonington Gray color to my sister in law and she loves it. She used it in a bathroom with a black and white floor and dark gray vanity. Don't have a picture but it looks great. Another gray that has a great review is Sherwin William's Collanade Gray. Check out this decorators review of it: Collonade Gray

PPG does not have a decent equivalent for SW 6119 Antique White. Chris is right, the color can be matched in PPG paints - they should be able to match this one pretty well, probably spot-on.

And I also caution you to evaluate your paint with all samples of all hard elements together - so any wood, stone, tile, carpet, major textiles like sofas, bedding, etc ALL together at the same time...again, pure white is great for cabinets and trim but something soft and not so bright for the walls will look more pulled together and ‘finished’. Your first inspiration pic where all walls and trim in all one color is not a bright white like Pure White. Your second inspiration pic which shows only trim/wainscot is probably a brighter white like Pure White however then I ask what was the wall color used with it and with what other fixed materials? If you want a complete white on white look like your first pic you will need to go darker than Pure White (IMO and Maria Killams:)