Does converting to JPG lose quality?
There is one lossy re-encode, but at the default quality it is very hard to see. The change you are more likely to notice is structural: HEIC is 10-bit and JPG is 8-bit, so smooth gradients in very bright or very dark areas have fewer steps to work with.
What happens to a Live Photo?
You get the still frame. A Live Photo is a HEIC image paired with a short video track, and JPG has nowhere to put the video. If you need the motion, export it as a video from the Photos app instead.
Why is the JPG bigger than the HEIC was?
That is expected. HEVC compresses far more efficiently than JPG, so the same photo typically lands 1.5 to 2 times larger as a JPG. You are paying file size for the ability to open it anywhere.
Are my photos uploaded anywhere?
No. The HEIC decoder runs in your browser, so the photos never leave your device. That matters more than usual here, since iPhone photos routinely carry location data in their EXIF.