Grace, A SIM card is your connection to the phone network. Think of it like the socket you plug your phone into at home. Which socket you plug it into (yours or your neighbours) decides what the phone number is, and who bills you. Same for a SIM card.
I can purchase a local sim card that will allow me to take photos, text and phone calls locally.
Yes, except you don't need a SIM card to take photos (or anything else that doesn't need a network connection), just to send them to somebody.
My apt has Wi-Fi capabilities and free Intl phone service. Therefore, I should be able to connect to the apt WiFi with my smart phone, correct??? I will then be able to send email back to us along with photos that I have taken, correct?
If you are connected via WiFi, you don't need a SIM card.
If you are on WiFi, you can send e-mails (and other internet stuff that needs a network connection), without it using any of your SIM card's data allowance.
If you are not connected to WiFi (outside the apartment), then all e-mails are sent via the SIM card and count against your data allowance.
"free Intl phone service" - I question that. The phone may have free calls within France, or even some free minutes to other European Countries, but I have never heard of free unlimited calls to any country.