Can a baby get a cold sore from its mother without direct contact with a cold sore?

Doubtful. Fever blisters/cold sores are caused by herpes simplex type 1 (not type 2!). The first episode is usually in childhood, and is transmitted by close personal contact, usually via secretions (like saliva) from another child or adult. Because the virus is inactivated by drying or by being at room temperature, it's unlikely to be spread by sneezing or by using the same tableware.
Very unlikely. Infants rarely develop so called cold sore, typically a reactivation of human herpes type i virus. Primary infection in infants/young children often presents as gingivostomatitis-sores involving gums, tongue, and uncommonly soft palate.