Custom shipping cost
Yes this is definitely possible.
We don't use any of Commerce's in-built shipping stuff - fields or calculation, as it's too limited for our needs (not Commerce's fault - we have very complicated shipping requirements!).
You will want to write yourself a custom module with either a full custom shipping calculator, or an adjuster for shipping.
You'd use an adjuster if you don't need to present any options to the user, and a full shipping calculator if you do.  If the latter, as long as you implement the interfaces for Rules and Methods, you can do just about anything you wants and present it to the user any way you want.  Can be a little tough to get going, but is very flexible once you've got it working.
Docs for this are here:
https://craftcms.com/docs/commerce/3.x/shipping-methods.html