For example, the order confirmation e-mail might say "Good news Joe, your order is on its way" while the apology e-mail might say "Dear Mr Biden, we're sorry but your order isn't on its way" and the parcel might say "Joe Biden, White House, Washington DC"
Should you ask the customer to fill out three different fields?
I'd stay away from gender/greeting too. Mr/Ms/Mrs/Dr etc. are way too culturally narrow even for the US.
Speaking of web forms and DC, I hate when they have "Washington, DC" listed as the state. Never mind that it isn't a state; at least the territories have to put up with that, so we're not alone. But DC should be in the Ds, not the Ws. Washington is a city, not a state(-like thing), even if it isn't distinct from the district anymore. Well, Washington is a state, but… you know what I mean.
Having had some pretty irritating experiences with systems that would only retain or use a piece of the names I gave them (either a family name or given name), resulting in a collision, whenever I feel that is what might happen I give them a concatenation of my names in different orders for the different fields, so if I was John Doe I might enter "Doejohn Johndoe". Apparently I'm not the only one doing this.
Yes, it often is. Years ago in college I worked in an international retail establishment. We were required by policy to address customers by their first names. It was a rural community with many older people, and many people were visibly unhappy being called their first name by a 19 year old kid they didn't know. And as an adult, I find myself in agreement to this day. If I don't know you, why are you calling me by my first name?
(If this is a burden on your users, you should consider ways to drop this requirement.)