Yes, those statements can be logged
Tools -> Options -> General -> "Log all SQL statements as INFO messages".
Additionally every SQL editor keeps a history of its content, which is persisted in the connections's workspace. You can scroll through the editor content through e.g. "SQL -> Editor history back" (similar to the history in a browser).
As this is stored in the profile's workspace, this history is retained over a restart (or switch of a connection/profile with a different workspace).
You can configure the size of the editor's history in "Tools -> Options -> Editor -> History size"
And finally there is the "WbHistory" command that shows you the list of statements executed in the current editor since the connection was established.
This history is not retained across a restart.
In my experience, a huge number of open tabs is a sign, that you should really look into using workspaces (see:
https://www.sql-workbench.eu/manual/workspace-usage.html )
I recommend to have one workspace per connection profile.
I also keep many workspaces around, that are not directly related to a specific profile, but a specific task (or set of tasks). I then load that workspace using "Workspace -> Load workspace". The 10 recently loaded workspaces (through that menu) will be kept in the "Recent Workspaces" menu, so if you have workspaces that you need frequently, they are easy to reach.
Regards
Thomas