Why do you need an outside border? Maybe get cobblestone or half size pavers if you feel you have to have something?

I don’t necessarily need one. But there is a difference in height between where the grass starts and patio begins and it kind of also acts like a retaining wall keeping the gravel underneath the slate in place.

You DO need a border or edging. Otherwise, the underlayment/paver base will migrate out and the slates will sink and shift. You need a commercial grade metal edging intended for paver restraint. You should be able to find this at HD or Lowe's or any landscape supply outlet. Quite easy to install when you are relaying the pavers/slate and virtually invisible once in place

Are you planning a DIY project? What are the dimensions? Quite a bit of physical labor required, possibly obtain a professional estimate too

For the outside edge you’ll need some retaining wall stone block. Visit a landscaping lot and see what is available that will compliment your slate.

Regarding your former hot tub site, to level everything must be removed and the base lowered to that of the larger patio. Reset the slate knowing some pices will require cutting and some new purchased due removal of the timber.

You're sure there is gravel? It doesn't appear to have a proper compacted base.

Belgium blocks/cobblestone make good edge restraints & risers.