Is the second usage understood or used in the US and Canada? I just heard it on a British drama. “How are you fixed for tonight?” “I’ve got something on.”

No, we would say "I've got something *going* on".

In the US, I don't think I would have understood it without context. I might have thought it meant "are you ok for tonight?" as if it meant I had a ride home, or something like that.

USA, I would never say that but in context i think it wouldn’t be hard to figure out what a Brit meant when they said that. For this usage we’d say *going on.* Is anything going on tonight?

US speaker, I’d never phrase anything the first way of that definition, but we do use “on” this way sometimes. “I’ve got a thing on Friday, can’t make it to your party.” “I’ve got a party on the weekend, you should come.” I don’t perceive it as particularly British either tbh.

"I've got something on" in the U.S. would mean: Do you need time to get ready? "I've got something (clothes) on." OR Do you want to watch the game? "I've got something (a show/movie/etc.) on \[the TV\]." OR "Something smells good." "I've got something (food) on \[the stove\]." But, most of the time, people in the U.S. would say the full sentence. And, no, we don't use it that way. We'd say, "I've got something *going* on." OR "I'm busy."

"Do you have anything on this week" would sound like "Do you have clothes on this week" but the second sentence would be understood.

for the second usage, in the US the question itself “Do you have something on (“happening”) this week?” would never be asked. you would ask “Do you have something *going* on this week?” or just “Are you busy this week?” as for the answer, in the US we would use “on” as a preposition to mean “the event falls on this calendar day.” in the UK they are using “on” as a verb to mean “going on” or “happening.”

No.