Vladimir,
"External SIGSEGV" usually means that the object you are trying to access is not created yet. So check where you create SQLite3Connection. (what form and when)
DataModules are good places to keep these things and also separate database code from the rest of the program. I usually put my Connection (named Conn) and Transaction (named TX) in a DataModule (named DM) and call:
DM.Conn.DatabaseName:= 'somedb';
If you do that, I suggest go to project options -> Forms and move the datamodule up in the creation-list so it is created first of all. Then you can access all the datamodule objects from any form that has your datamodule in its uses clause.