1. Garden project: it is based on alternating wood floor and flowerbeds.
The wood floor allows you, your kids and your dog to walk all around the garden freely, nearly as much as if you still had grass.
Moreover the wood floor is in harmony with your house style, it is easy to lay and it doesn’t seal up the ground, this is very important when it rains in downpour; I am not sure but maybe laying down the wood floor also by the side of your pool could solve the dirt problem.
The flowerbeds let you enjoy a garden full of colours and life.
The plants are perennials and a shrubs (Buddleja), they flower at different times, so you’ll have flowers from April till October, they also need very little maintenance and very little water (but remember that the plants need more water in the first two years till they develop the root system and became less vulnerable to possible drought).
The flowerbeds alternate surfaces with plants and pebbles, this was a request of yours but it also helps to achieve two important objects: to give the necessary movement to the garden design and to limit water transpiration and weed growth.

2. Kid’s games: the sand playground makes possible many games, even at different ages and growing up the kids can still find ways to enjoy themselves and play together.
It is also very easy to replace it with something else, wood floor or flowerbeds, as the kids becoming teenager won’t need it or like it any more.
The position is thought to let you have a free direct view from every seating – relaxing area because till the kids are very little it is nice to keep an eye on them sometimes, the perennials all around have different heights but never so high to cover the view.
In addition to sand playground, the kids can run all around and also (tri)cycle.

3. Dog’s area: I placed this area in the remotest part of your garden so your dog can have her/his privacy and you as well.
The flowerbed between sofa and dog’s area has the Buddleja shrubs that grow big and high enough to create a natural fence.
As for the material I think pebbles is a good solution, anyway I advice against grass because it is a high maintenance solution that doesn’t make sense for so small a surface.

4. Seating area: I placed it where you can have a full view on your garden and so enjoy all the colours, flowers and the growing of your plants, which is an enchanting view.
The sofa and small table that I advice are white, because it gives much light to the garden and does not compete with the garden’s colours.

A. Sofa
Options vary according to number of seats and prices, the style is quite similar
I think white is the right colour, Ikea sofa unfortunately is only brown but the price is so profitable that I could not omit it
A1. www.unopiu.it, Unopiù, Agorà, 3 seats sofa, white: € 2.850
A2. www.unopiu.it, Unopiù, Sunlace, 5 seats corner sofa, white: € 4.110
A3. www.ikea.com, Ikea, Arholma, 5 seats corner sofa, brown: $ 670

B. Table
Options vary according to the respective sofa series
B1. www.unopiu.it, Unopiù, Agorà, small table, white: € 485
B2. www.unopiu.it, Unopiù, Sunlace, small table, white: € 295
B3. www.ikea.com, Ikea, Arholma, small table, brown: $ 49,90

C. Sun loungers
I think sun loungers will be useful and nice, the first option/photo is the ones that I really love but as they are very expensive and delicate maybe you could postpone this purchase when the children are grown up. So the second option/photo shows an alternative that I find both nice and practical because it is easy to store and move where you want and need, from wood floor to pool and back, and also lets your kids to run all around more freely
C1. www.unopiu.it, Unopiù, Swing, sun loungers : € 1.300
C2. www.unopiu.it, Unopiù, Roger, adjustable back sun loungers: € 205

§. Armchairs and table
Eventually I advice white armchairs and a table for the Tiki hut, the chairs could be useful also to add more seats if needed around the sofa area
§1. www.ikea.com, Ikea, Angso, armchair, white: $ 69,90
§2. www.ikea.com, Ikea, Angso, table, white: $ 139