Is there some way to read/write table properties directly by SQL or some script?

Presently I can view table properties by right-clicking a table. If the table is located in some folder different from the project root the table properties window will show the folder location .

Yes,

ComponentProperty for SQL in

https://manifold.net/doc/mfd9/other_sql_functions.htm

And Database.GetProperty in scripts

https://manifold.net/doc/api/scripts-net.html#database_getproperty