The standard Windows installation has already a large number of fonts, more than you usually need.
I would recommend to only add those fonts you really need e.g. publications and titles etc.
1) The fonts that may be required for PD titles will be recognized by PD. As you have seen, you'll get a message and offer to download the required font. These fonts are downloaded and stored in the directory:
C:\users\public\cyberlink\Downloaded font
2) The fonts downloaded via the Application Manager are stored in the c:\windows\fonts folder (file name is a number), where fonts should be.
When you list the installed windows (-11) fonts via the control panel you will notice that for the 1) situation, PD fonts there is a small blue arrow in the corner, indicating that the font is somewhere else.
Again, unless you are a font collector, it is good practice to have only those fonts installed that you really need.
I always wonder why CL makes such a large font collection available. They do the same with stickers and stuff.
It would be good to scrutinize your font collection and uninstall those you don’t need, as a large collection of fonts might have influence on the performance of your system, but only when the font set is really large, and/or the system is of lower capacity.
You mention that PD hangs when it tries to download a 1) font. Maybe a smaller collection of fonts will already do the trick. You might want to look at the above give font locations to see whether they exist and can be reached by PD (a protected folder or the like).
You may want to share your system information with us here, by making the DXDiag.txt file available.

But, at the end of the day, when nothing is wrong with your system, PD should be able to download fonts without any problem. It is always good to notify CL customer services about your problem. Be prepared that they will ask all sorts of things and suggest to re-install PD, and check your windows installation using the SFC and DISM command.