If it's right after drinking that she burps, it's more likely just air she swallowed in the process. They tend to do that, which is why you're not supposed to let big dogs drink forever because that swallowed air can cause bloat.
The hiccups... Still a mystery. I think it's in some way linked to diabetes, which freaks me out (this is just MY hypothesis, nothing ever proven scientifically) because Jemma gets them all the time.