I can't address the Firefox issue, but perhaps can help re iTunes and iPhoto.
Do you have an external hard drive that is large enough to hold your "iTunes" folder (in your old home folder) and your "iPhoto Library" file (also in your old home folder)?
A USB flash drive might also work.
(I can't help with time machine, I have never used it so I don't know if this could be done from a tm backup drive)
What I would do:
a. Connect the USB drive to my older Mac.
b. Go into the home folder and copy the iTunes folder and iPhoto file (it's really a folder, but just copy the icon) to the USB drive
c. Disconnect the USB drive from the old Mac and connect it to the new Mac
d. I would open Photos on the new Mac, and choose to "import" the entire iPhoto library. There are other ways to do it, but I'd try this first. Does it work for you?
(if you start getting "permissions errors", there are ways to fix that)
Regarding the music...
I don't have any interest in the new "Music" app, because the only thing I used iTunes for is the free streaming internet radio player. However, you might want to use Music, to see if it can import from your older iTunes folder.
Another way to do this:
Do you know that you can STILL KEEP USING iTUNES, even on a new Mac (that supposedly doesn't "support it" any more)??
There is a free utility out there called "Retroactive" that will install a working copy of iTunes onto new Macs.
Then you can keep using iTunes as you used it before.
Find Retroactive here:
Retroactive has been discontinued. You should transition from Retroactive to supported apps such as Music, iTunes for Windows, iMazing, Photos, Darktable, Lightroom Classic, and DaVinci Resolve. - ...
github.com
(scroll down to find the "click to download" button)
I'm thinking that if Music won't do what you need, you could install iTunes and then try importing from the USB drive.
Again, there are other ways to do this, as well.
Good luck.