Charles, here's the order of operations

Root flares

This is first on the list, especially on the ficus. Gently dig back the soil around the trunk until the flare, where the trunk widens into the roots, is visible above grade.

Here's a video walking through it: Watch on YouTube

If you have questions once you're out there, just let me know.

Turf along the back patio

Keep this section as is. With the pavers running through the turf, converting that strip to real grass would be more trouble than it's worth.

Putting green area

This one, on the other hand, converts easily. Putting green turf is usually installed over ground that's been compacted hard, so before any sod goes down, have that ground tilled, broken up, and topped with compost.

For the grass itself, go with St. Augustine. It reads more tropical and handles shade better than bermuda would in that spot. Picture below is my own St. Augustine, so you can see what it looks like established.

Vine for the dead olive tree

My go-to here is tangerine cross vine. You'll need to train it up the tree and help it get started. Tying some wire to the trunk gives it something to grab onto early on.

Other options if you want to compare: coral vine and pink bower vine.

Coral vine

Pink bower vine

Back wall, pond feature

Same vine options work here, tangerine cross, coral vine, pink bower, plus one more: lady banks rose.