Only data encrypted in cookies ever leaves the server
Do you edit the .env or config/session.php directly?
Of course
What is the most secure session driver? (file, cookie, database, apc, memcached, redis) etc.
Secure in what way? All drivers are secure from external influence. They have other pros and cons, but security is not one of them. If you are concerned that the client might be able to decrypt the cookie based session, then don't use it.
Will encrypt session encrypt anything that is added to the session via put('key', 'value');?
yes if its enabled
Is strict the highest security level for Same-Site cookies?
Its the most restrictive regarding when the cookie will be sent to the server. It changes nothing about the security of the cookie itself
I am using a API that validates user when provided a username
Then if you are using an API then you will be in a stateless session, ie NO session, so NO session storage