To answer your question in the title, UA is required to give you nothing. Even for cancelations/delays that are in their control, they aren't required to, though they usually do for longer delays or cancellations.

It sounds per your post like after your flight cancellation, rather then take their rebooking 3 days later, you asked them to route you via CUN with someone else operating the last leg, and they couldn't. It is very possible, I'd say likely, they don't have an interline agreement to re-ticket you on the carrier doing the last leg, but I think there are others who are better able to look up if it would have been possible.

After that, it sounds like you asked them to change the destination to CUN rather then Belize, and you booked a one-way on your own So they basically, because of the weather and at your request, changed your destination without charging you any fees. Great. But when you asked them to do that, their contractual obligation changed to get you to CUN, not to Belize. When you make booking arrangements on your own, at your own discretion, it's very unlikely that they will refund you that amount, and in particular, if you changed your destination. I

f you wanted to them to get you to Belize, I would have suggested other routings that you looked up - there may have been something they could have done (does AA fly via DFW?). I realize this is all in hindsight now, but if this kind of thing ever happens again...

As to the pre-paid reeservation, carriers will almost never reimburse for that. I have had hotels that have given refunds due to canceled flights or weather - but those were always big chains - I can imagine an independent would be less willing to budge.

If you have travel insurance, forget fighting with the travel providers and just make a claim - it was a flight cancelation. They should cover it, though they may ask questions about why you bought a new ticket from CUN vs. your origin or connection city.