Best settings to use with W3 Total Cache

Hello there,
If I use the cookies for storage, will it work with w3 cache plugin? Is there an option for me to change in W3 settings?
Also if I use the Session storage, REST API completely breakdown. You can see it in Site Health Check plugin. Loopback requests also stops working.
As far as I can tell, cookie storage causes redirect loops, for my setup.
Sessions breaks API.

Also I can’t see any $_GET parameters added to the url’s even though I haven’t enabled JS option.
Can you please point out the correct way to use this plugin with W3 cache and Cloudflare enabled.
Thanks.
Site: paradise.carpetar.com
Status info added to access info for security.