Honestly I highly, highly doubt it has anything to do with the microchip. So much so that I wouldn't even fret about your other questions unless it is checked out - it will be easy enough to tell at her appointment when they scan her.
You need to call your vet and explain what you think has happened. That lump needs to be looked at, whether it's the chip or not. Explain the vomitting and any other signs, like lethergy, diahrrea, no appetite, etc. Swelling and pain are two indications of infection or inflammation. The other two signs of infection are redness and pus.
Biscuit's microchip (another brand) migrated to her armpit not too long after it was implanted but has stayed put since. I don't think it really matters if the chip migrates, as they're supposed to scan the whole body anyway, but the swelling and pain are concerning. Whether or not it's microchip related, I agree with taking her to the vet.
One of my dogs home again chips moved, but it stays right up around the shoulder blade area and its not that big, feels like a small grain of rice that is moveable. Our vet left it, since they pretty much scan the whole body anyways. You can probably contact homeagain and they can put it in their files that the chip has moved ( if the lump is the chip) and if you ever report your dog lost make sure to put on your fliers that the chip has moved.
Doubt it's the chip. If anything, gravity would draw it down towards the front leg/armpit. Sounds like maybe an abcess from a shot given in the hind end, that gravity has made drop down hind leg. Or if you own a cat, a cat bite. Or spider bite.
I would be concerned because even though they are "supposed to" scan the whole body they don't all do that. The Humane Society I worked for would only scan between the shoulder blades. They were always in a hurry to process the dogs/cats and move on to the next thing.
Highly doubt as well that the microchip would have migrated that far.
Bella once had a "knot" like that from one of her shots - a "sterile abscess" they called it. She has also been bitten and stung my deerflies, skeeters etc that leave bumps on her. Could be a reaction to a bite - even a tick.
Let us know what the vet says. Hoping she is all better real soon!