Do you have mercury amalgams? These can give off vapour and thus lead to mercury intoxication. A mercury-intoxicated mother can pass mercury to the foetus through the placenta.
Your daughter definitely sounds as though she has a very compromised immune system (given the susceptibility to infections and pathogens) and, as you say, compromised liver function. These are both possible indicators of a systemic toxin such as mercury, although mercury isn't the only possibility. If you can feedback, then I may be able to offer further appropriate suggestions.
That your son is also exhibiting similar propensities also lends some weight to the above heavy-metals hypothesis.
Alternatively, do you have mold in your house - e.g. in the children's bedroom? Continual exposure to mold can be challenging for the immune system to have to continually cope with.
Whilst intrahepatic stones and/or Gallstones may be a possible cause of your daughter's liver issues, there may be other causal factors such as the aforementioned. The more symptoms you notice and document, the better your chances of building an accurate hypothesis.
As far as the reflux is concerned, this might be due to insufficient stomach acid (which, of course, can be an issue when H-pylori is or has been present, and might also manifest in the absence of sufficient dietary zinc). Alternatively, there may be some bile refluxing into the stomach via the pyloric sphincter. I seem to recall that H-pylori can mess with the pyloric sphincter but I'm rusty on H-pylori as a subject area. David Hompes is something of an expert on all things H-pylori, if you happen to be seeking a practitioner who knows about that. Nice, sincere guy, too, and charges reasonable rates.