I wouldn't recommend white washing the pavers. They won't last.

If you can put doors on the whole back area and deck or pave out to a functional size for entertaining - be sure to have as close to same height as inside and the whole width of the house if possible. You could do the pergola which looks gorgeous and makes a cozy feel. On top of the pergola you may be able to do a weatherproof retractable awning if you do not want a permanent covered area. That would make a whole area for indoor outdoor entertaining in all types of weather. You can still have a step down area to grass and plants at the rear. Travetine is great and easy to hose down. You do not have to concrete them down use pavers on a good road base. You would have to speak to the experts about how to get the right height - also be sure to have a slight fall away from the house.

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You could build a deck over the paver which would be the easiest, pave over existing pavers but this would require levelling to have a perfectly flat surface, there are so many different products on the market for diy or you could use a tile paint, but decking is an easier option.