On your first question, no the append query is not supposed to "move" data or insert to the new table and then delete from the old one. Append query inserts data to the new table.
If you want to delete data from the old table, then you need a delete query. When using a delete query make sure that your criteria are correct.
For your second question: Yes, it is possible to run more than 1 queries on button click. You have to create a macro and use OpenQuery action (with the appropriate parameters) as many times as the queries you want to run. Then assign that macro to run on button click.
Hint: You might want to use the SetWarnings action to disable warnings before your queries and re-enable them once you are done. This won't prompt the user about the number of records affected and if they want to continue or not.