[SalesForce] Sharing of a junction object

No. Sharing rules open up access to records. If the user's profile does not have edit on the contact object, he will not be able to edit the record.

What does "Read/Write" access on Sharing Rule really signify if an Object has only "Read" access on the Profile and that the record can actually never be edited?

You'll want to read Sharing Considerations for most of the usual complications of sharing.

However, for the purpose of this question, it's important to make a distinction. A feature that requires a certain Sharing level of access to the record does not evaluate Profile Permissions regarding that feature. You can have Edit access to a record without having the Edit Object Profile Permission for that object.

If you have Edit access to an Opportunity, you can do all of the Edit-level features to that Opportunity, even if they can't edit the Opportunity itself because of Profile Permissions.

For example, the user can add Notes and Attachments, add Opportunity Team Members, Contact Roles, and so on, even if the Profile Permissions don't allow editing any Opportunity.