Most importantly would be that they're just about absolutely necessary these days for playablility on emulators, and that they're somewhat unique in that way since many other systems don't have a need for them.
NES headers are crutial in order to know what cartridge hardware to emulate. This is hardly unique. SNES have several carts with different hardware. Emulators can (and usually does) use heuristics on the ROM to determine whether they have a LoROM or HiROM cartridge, and Nintendo had an imposed header in the ROM so emus uses that. But there's no guarantee the header in the ROM is correct, actually you could even have something completely wrong in the ROM header or remove it entierely and have a game that works on hardware - but it wouldn't work in emulators. How hironic.
Many NES ROMs have also a header in the ROM but the format is less standard, and info is largely incomplete and often plain wrong, or missing entierely (about half of games). So this header cannot normally be used to emulate the system.
So that brings me to the FDS games. What's the nature of headers for that system?
FDS headers are mostly useless and contain no information. Basically their only purpose is to verify that the data following it is effectively a FDS dump. Info about disk sides amount, etc... is found on the disk itself in it's internal header. Which is itself used by the FDS BIOS.
I've noticed a few things while looking at instructions and trying to find correct versions for the games. Some specifically say that the rom MUST be unheadered for the patch. Most don't mention this. Some show a pre-patched value of a headerless rom on the page. Most don't.
So basically it's the same hell as for the SNES
Should I be stripping all of the headers before patching these games, or just the ones that specifically say no header?
I think if they don't say anything it's with header ? In all cases you should try both and see which one works (like with the SNES), unfortunately there's no real alternative.
Are the headers necessary for the FDS?
Will I have to add my own headers somehow after patching non-headered games?
If an emulator you wish to use only supports headered FDS dumps then yes. I have no idea which emu supports or don't support headerless FDS dumps, but it seems with header is the "standard".