What are the possible meanings/connotations of this?
Is it only romantic or can it refer to time and scheduling: "Dzisiaj, nie mam czasu. Jutro, jestem cały twój"
I use it a lot when switching attention from something I was doing to a person, for example when using computer and someone asks me something.
Your example is also OK, I can see myself using it like that, although rather with close friends than, say, work colleague/acquittance.
But note that this is somehow person dependant, one of my friends find it odd I use this phrase like that.
Can you just say "cały twój" for short? Like how you can say "all yours" in english.
Not as a translation here. But as an answer to something, maybe it could make some sense.
How does gender change this? If i were a girl would i say: "jestem cała twój"? Or boy talking to a girl "jestem cały twoja"?
The gender of the spaker affects both the adjective (cały) and the pronoun (twój). So a boy will say "Jestem cały twój" (regardless on whom he's speaking to) and a girl will say "Jestem cała twoja".
Well, that definitely looks wrong in English. "completely" would work though.