I have setup an approval process and set record editability to Administrator or Current Approver.
When I submit a record for approval will it be automatically be shared with the approver? What if the assigned approver doesn't have access to the record? Can the approver still edit the record?
The assigned approver requires edit access to the record to edit it. The approval assignment will not 'upgrade' the authorizations of the approver from e.g. read-only to edit.
From the docs (https://help.salesforce.com/apex/HTViewHelpDoc?id=approvals_creating_approval_processes.htm#Approver):
The assigned approver must also have edit access to the record through both their permissions and the organization-wide sharing defaults for the given object.
