Does The Things Network (TTN) have a feature to store data and maintain data logs even if I shut down my laptop or close the TTN console tab? The gateway and node will remain active, of course.
Yes and No
You should have an integration setup to keep your data - this is fundamental to an IoT setup.
As a measure of last resort there is the data storage integration that will keep uplinks for about 24 hours but it is not designed to be used as your primary database.
Given the integration is actually called Data Storage, I think you might want to try searching the documentation and/or having a look around to see what’s available.