No there is no OOTB feature to get this report. You can export your logs to an external service using one of the Export Log to X service extensions and filter them yourselves.
You can get Monthly Active Users via Quota utilization report from Support Center (login to auth0 dashboard and click on Help & Support link at the top and then go to Reports section. Note that, this report only give you monthly count of Active Users where active users are defined as users who have logged in at least once in last 30 days. So for eg. if someone logged in only once in last 30 days they are counted as 1 active user and if someone logged in 30 times in last 30 days, they are also still counted as 1 active users (it’s the same user logging in multiple times)
It depends if users autologin after signup. The default Lock behavior for database users is that they are automatically logged in after signup. So yes, after signup they will be counted as an active user for next 30 days. You can turn off that behavior if you are using lock though Lock Configuration Options