The problem as I see it is that this network drive is showing up in Library's which isn't the correct place for it to show up as you aren't using Homegroups and therefore not using Library's. It's no wonder things are messed up.
It looks like you added this drive to your Library's when you had the Homegroup set up. You will want to remove
the drive from the Library and remap it. Although it looks like you already have the drive mapped because it's showing up under "Network". Map Network Drive
Also besides just checking the boxes on the sharing permissions
tab of this drive you will also want to use the Security tab
, click "Edit" then next window click "Add" then type in "Everyone" click OK, then second picture notice that "Everyone" is on the access list now, put check marks for full access, hit apply, OK. Second Picture below. This will allow access for this drive to everyone in your network.