Not sure what your budget is, but how about a Pergola? It would look absolutely beautiful, doesn't cost a bomb to install and you could add some fast growing vines, be they flower or grape, or a mixture of both, whip the slabs up, lay a plastic membrane and then whack a couple of tons of peashingle down. To finish it off, I would remove the wall in front of the patio and replace it with some large planters containing box hedging, rosemary, or lavender. A little touch of the continent, right on your backdoor step and you won't even need to go through passport control!

7m x7m precludes palms really and don't forget Date Palms and Washingtonia are both affected by the Palm weevil and will need regular treatment. If you don't live there full time and are unable to get regular maintenance carried out, go for indigenous succulents and things like grape vine ( I have 5 huge vines at my house on the mainland, they make beautiful shade in the Summer, produce kilos of grapes and the only care they get at the moment is a trim in the winter and a thorough watering in the Spring as I am only there twice a year until I retire there next year). I like the look of Bougainvillia but it can tend to overgrow a small space and is unpleasant to trim back with its spiteful thorns, Wisteria is a better bet for a climber but takes a while to establish and produce flowers. Lantana, Plumbago and geraniums are all indigenous and do quite well without too much care, there are also some dwarf bamboos that do well in hot dry conditions, common Ivy does well here but is invasive. I know that there are a few good garden centres around the outskirts of Palma so go and have a look around,many of the labels give you good information about habitat and growing conditions and I am sure the staff would be only too happy to give advice. I have lived on Ibiza for the last 13 years, it is a little more humid than some parts of Mallorca but if your place is to the West of the island closer to the mountains that will also be fairly humid, there are plants there that have evolved to survive on the moisture from the nightly dew. There are also a lot of ornamental grasses that will grow with a little irrigation, some of them change to fantastic reds and browns toward the end of the Summer and produce beautiful seed heads, a range of Ryes, Timothys and fescues would give a layered effect. A late thought, looking at your photo, if you have trickle irrigation you can install some of the pockets for a vertical garden on one or more walls of the yard to produce a lush, cosy atmosphere. The pic is a Portugese villa from Google images.

It’s a jolly nice garden that just needs a few tweaks. Is the washing line retractable, if not, then swop it for one that is. That way it is tucked away against a wall or fence when not in use. Most supermarkets sell them as well as Dunelm etc. You have some wooden raised beds filled with wood chips which are crying out for low growing plants such as lavender, geraniums and aubretia. Your pots look a bit of a trip hazard as they are, so they would be better grouped together perhaps on the right of the steps? The best fence treatment is plants - climbing shrubs like honeysuckle, roses and jasmine are fast growing and stunning. Clematis are quite delicate so better grown up an obelisk or through other shrubs such as roses. My own garden is quite small and just a rectangle of sandstone paving, but I’ve surrounded it with plants to soften the edges. Here’s a couple of pics - one of today (raining!) and one last summer.

I am a retired garden designer. Sorry I don’t have time to read all the comments, so apologies if I am repeating what others have said. I think your plans show that you have thought this out and worked hard. I would n’t have a brick arch. It will look heavy. I can’t gauge the scale, but your borders look as if they may be under a metre wide in some places. If a border is 60 cm wide you will probably get in a climber at the back and a perennial layer at the front. It won’t look ....substantial but may just about fill the bill. It seems as if something less than this is what you will look out on from your patio. The border around the edge of the lawn tapers to a point. Generally speaking there isn’t really a plant that you can put there and expect to form that shape. Sorry that is all I can remember without referring back to the drawing which doesn’t seem possible without posting or deleting! Good luck and have fun.