I have western red cedar shutters and I went dulux antique white. Painted on the wall and ceiling.

I would wait for the rug, until you have the shutters up and your furniture placed..it may take on a character of its own and you'll want to compliment. My first thought though was to coordinate with your kitchen and incorporate some black, with a black/beige stripped rug, which would be fun with the wallpaper and perhaps paint your chairs black, add the shutters and you could paint the window trim black and add a large black chandelier.

I would clean up first. Especially the stuff growing in the paving. Then get gardening with pot plants and start experimenting with different plants and the positions. See what you like and if you can keep it alive! Our house has the sun setting over the back fence and we've done a few inexpensive things to reduce that issue (most expensive were outdoor blinds around the verandah) but for most of the year we just don't want to be out there. It is hard work to keep the plants going in summer, there is a lot of sand due to the difficulty in keeping lawn alive and it is just bloody hot. So, that's why I'm saying clean up, start gardening small and see how you go before committing to structures that might not address the real issues you have with the yard. Just my 2 cents! :)

You can do whatever you like but there needs to be unifying factor such as colour and a cleanness of line. It is important any trims match in material and colour. Best to go with painted shutters. Bear in mind they will be seen from the road so think about any other windows seen from the same position. I have successfully mixed shutters and a drape. Shutters are expensive so get a professional opinion and measure - I went with the top local retailer and they still messed them up and i had a three month wait to get them replaced. All the best.