My bad,.. I must have incorrectly specified the database host during the install. My .env file said the DB_HOST=127.0.0.1
and I needed to set it to DB_HOST=mysql
I might mention that I am running on Docker containers using Laradock.
With that change, I am now successfully connecting.
As far as PHP7 and PHP7.1 are concerned I can confirm that AsgardCMS v2 is running fine on PHP7 and PHP7.1 (I'm using PHP7.1 myself for a project).
For AsgardCMS v1 it should be running fine on PHP7 and most likely on PHP7.1 also, but I have not used v1 in a long time, so I can't tell you 100%.