In some cases, the difference between correct and incorrect can feel aribtrary. But here, what would be the status of ‘visa’ in “Je fais une demande un visa”, another direct object?

Note that you can also say “Je fais la demande d’un visa”.

Often, we say “Je fais une demande de visa” because say “Je fais la demande d’un visa” tell that you can have more than one.
Oh, yes you can, but the middle french have just one, or not at all.
However, “une demande” is different of “la demande” because the visa is limited in the time, you can request a visa several times.

But, there is no perfect answer, all french people understand “Je fais la demande d’un visa” 😉

Je fais une demande un visa

This is wrong and makes no sense for me. You could say Je fais une demande d’un visa but it sounds strange to me, because it would means you want a particular visa and not anyone, and you don’t specify it in the sentence. So you could say Je fais une demande d’un visa Schengen for instance, what looks less strange to me.

No you can’t.

And it is “une demande de visa”