Where are you located (nearest big city & state) and how old are your children? Have you considered mulching that area while the kids are young then planting it in a few years?

" Should I remove that small wall that extends from the hot tub before having new concrete poured or keep it? "

What small wall? What is it's purpose? I think that is it in the corner of the pic but cannot tell what or what it is there.

" Should I go all the way to the house with the new concrete or put a garden space between the house and extension? "

How about showing the entire yard? It is thought to offer advice in a vacuum. What region of the country?

IMO concrete is not IMO ideal for kids play area ever. I think mckh or fake grass woud be a btter idea. Not to mention the trees need to get water and too much concrete usually kills the trees eventually Where do you live is question we need answered too. Why do you have your BBQ reight next to the spa? I think I need to see the whole yard and hous ein pictures here Post them in comments DO NOT strat another post.

Here are a few suggestions: