Getting feedback on Twitter is excellent for big picture questions (career pathways, publishing), but not so good for specific scientific ones like you were asking. I would have answered your questions, but I couldn't. I do not have the knowledge.

If you went to a general scientific meeting, like AAAS, and asked people these questions there, would you expect useful input? I think that's the sort of level where Twitter is at right now.

As a grad program coordinator, I've had some variation of this question several times.

It does not hurt to ask. That is what the program coordinator and the like are there to do

But expect the answer to be, "We won't review your application until you get us the GRE score." Typically, these are labelled "requirements" for applications. Offices have to be very careful about treating all applicants equally.

Incidentally, you might characterize the GRE as "some silly test," but I've seen some surveys that indicate most grad program consider it THE most important factor they consider in reviewing applications.

Cynical mode on:

Pedigree is code for "old boys network."

I am not a dog bred for show. I am a person who has entered a career path.

I wouldn't say postdocs are being lied to, but postdocs may be told truths that have a very limited perspective.

I suspect many faculty have experience of participating in job searches in one department at one institution. Not a good scenario for induction.