I installed it in to Windows rather than Inkscape and usually copy from working files (although I also know the correct format for font family.) This usually is not much of a problem as few people make parts thus there isn’t a lot of effort going it to documenting it. There are a couple of tutorials (that apply to current versions of Fritizng) here, one of mine and one from Old_Grey that may also help because part making is complex and bites without warning (as you have discovered!) You may have already seen them as I post this regularly when part questions come up. If the font is incorrect Fritzing will substitute one of the standard fonts in the sketch and write that out.
I lately learned there aren’t links to the videos in Old_Grey’s tutorial so you need to do a google search for the title and then they come up on YouTube.
It should not be unless you are saving the sketch as a .fzz file rather than the part as a .fzpz file. Fritzing will put all custom parts (any that are not in core parts) in the .fzz file and loading that will put them in the temp parts bin. The parts editor (if you are using it) will store them in the mine parts bin when saved as a new part. My earlier example that wrote to the mine parts bin was from the parts editor.
Peter