That a BBQ is a fire waiting to happen it needs open air for safety I even like an actual hood fan over them in covered areas . I kind of like the idea of a screened in area on a buggy day we just scamble for the house. What do you think you will use the space for? Where do the stairs go it seems to be a one storey house with no basement . How do you access the space form the house ? Maybe some more info I like the brick floors i you open this up with pillars all the floor will need to match so either brick for the patio or cancrete for the whole thing. It is kind of like when you cnavert a garage to a FR it will always look like a cinversion unless you get rid of the driveway completely .

I have made a boardwalk, about 12” wide. It is temporary, and easy to remove. I have also used; used wood pallets, that was not as easy to remove. The pallets were free, the boardwalk, a couple of hundred dollars. I reused some of the boards for a potting station/plant nursery. I dismantled the pallets and burned them. Maybe a couple of sheets of plywood to give you a porch/patio. Maybe $100.

IMO 9x9 is too small for any outdoor space I do think you need to wait until you are in the house before deciding how the outdoor spaces will actually be used. Houses have a way of telling you how they are used and often not what you eaxpected. I can’t imagine how the lower patio will be used for much since it requires a set of stairs to get from the kitchen to that space so maybe that would be a great space for some seating and a hot tub. I like multi level decking with large areas for cooking. dining and them much like a house the seating space on a different level It also make it easier to have heat control. So wait until you are in .These are a couple of inspiration pics to get your ideas flowing.

I think ipe deck tiles would match the ceiling really well, and provide some nice contrast with whatever you end up redoing the pool deck with: