Stock transfer question ...sort of

I own a few shares of stock.
Since I bought them as a single young man facing the draft I bought them as a joint tenant with my father as the other tenant.
They belong to me.My family has no question about ownership.
My father died recently and I would like to transfer the tenancy to my wife.
The stock investor service that is handling the transfer wants me to send them instructions.

1 What should I include in the instructions?

2 Why do they need the original stock certificates?

3 Hi Opal

4 Is this really a transfer?