All y'all are having a lot of fun at Houston's expense. Since I live in Houston, I feel compelled to defend it. However, most of what you're saying is true-ish, if not true outright.
... try and avoid saying "I have to shoot off"!
With a reasonable audience, you can get away with it - indeed, you can get away with almost anything - if you say it with a British accent. Teenagers and young men who don't date much will probably ridicule you for it, no matter what your accent. Unless you're from Yorkshire, in which case no one will understand you at all.
Never use "pop" or "soda".
I live in Houston now, and I can't remember the last time I heard generic soft drinks called "cokes." It's a ferocious melting pot here, and things change fast. I say "soda," and always get away with it. Out in the hinterland, though, everything seems to be a coke.
If he prefers iced tea sans sugar he must specifically request "unsweetened tea".
Yes, he must. The default state of iced tea is sweetened. In the broad swath of the south, an especially sweetened tea has come into use, made by saturating it with sugar while it's hot; that's called, "sweet tea," and you generally have to ask for it specifically. It the server asks you, "Sweet tea?" and you decline, you'll probably get sweetened tea.
"Fixin-to" = "planning" or "going to".
"Fixin' to" is used by the speaker to explain why he's idle at the moment. "Whatchoo doin' up there on the porch with that mint julep, Buford?" "I'm fixin' to go out back and dig me some post holes." Post-hole digging is a lot of work, so it sounds like the speaker is very busy, while fixin' to dig post holes is kind of relaxing, and may involve whiskey or beer.
... don't ask for a rubber.
Indeed. Even an upperclass London accent won't help you with that one. Especially, don't send your young daughter into the drugstore to ask for them.
Y'all is singular ...
Yeah, sort of. In my experience, the expected form of address - "you," "y'all," or "all y'all," - depends on the angle that the listeners occupy, from the speakers viewpoint. Narrow angle, "you;" up to about 90 degrees, "y'all;" more than that "all y'all."