There is no storage limit so your premise regarding fees fails so requires no further comment!
You can "Bulk Delete " emails if you want. There is no single button to do that but the following should help.
Click on the box to the left of the first email you want to delete and assuming you want to delete all the emails that are below that you can either press on your keyboard "shift" key and "PgDn" key repeatedly and it will mark all the other emails with a tick or use the "shift" key and the down arrow. This will also mark all the boxes with a tick or tick the first box you want to delete and holding the shift key move to the last email you want to delete and tick its box and release the shift key. All the emails in between should now be ticked.
If you don't all the emails that have been marked with a tick deleted you need to go down the list and untick the ones you want to keep.
Once you have done that you then click on the "Delete" button in the menu bar and if that doesn't work try and drag and drop them into the trash folder.
However why not delete the emails as you go along instead of using your Inbox as a storage folder which is not what it is designed for. If you can not make up your mind about whether to delete or not you should set up a "Pending Delete" folder or something similar and when you do decide everything in that folder can be deleted you just delete the folder in its entirety.
