While creating or editing category, you will get a field name called as Visible In Menu, please select yes here and provide position for it - https://prnt.sc/q2amkx
Yes of course there is a way to create your own shipping method in bagisto .
You can create your own package or you can check how to create shipping method.
We have already fire an event after product adding ( checkout.cart.add.after ) and removing ( checkout.cart.delete.after ), you can listen this event and can perform desired action.