There's a couple of ways you can check how many days have past since you installed the evaluation edition:
From Pinal Dave:
SELECT create_date AS 'SQL Server Installation Date'
, DATEADD(dd, 180, create_date) AS 'SQL Instance will expire on'
WHERE NAME = 'NT AUTHORITY\SYSTEM'
Alternative solution - Finding Your SQL Server Evaluation Time Limit:
EXEC sp_configure 'show advanced options', 1;
EXEC sp_configure 'Agent XPs', 1;
DECLARE @RemainingTime INT
DECLARE @InstanceName SYSNAME
SELECT @InstanceName = CONVERT(SYSNAME, SERVERPROPERTY('InstanceName'))
EXEC @RemainingTime = xp_qv '2715127595', @InstanceName
SELECT @RemainingTime 'Remaining evaluation days:'
The evaluation edition is a full edition without limitation (other than the timeframe you can use it for), as mentioned in the License section of the Docker hub.
I don't believe you need to enter a serial after you've purchased it (you just need to retain the purchase information Microsoft provides in case you were audited). You can follow this guide on how to upgrade to a non-evaluation edition.