Here is the tab for what I'm talking about:

F|-3-----3---3-----3-----3------3-----3-----3-----3-------3--------| X2

I have the F as my index finger, A as my pinky, and D as my ring finger and I can't do it. So is that wrong?
That's the easiest way to. Just make sure you have your thumb flat against the back of the neck, not curled over the top because that really lessens the amount you can reach with your fingers.
Seems normal enough to me. If it feels awkward try stretching your fingers maybe? It'll absolutely get more natural as you keep playing.
My sig? Nice.
i would think it would be easier for you to just bar the 3rd fret for 1st and 3rd strings, then you could just lift your ring finger on and off while keeping your pinky on 6
You need a 3 string barre the 3rd fret as was suggested.

I just tried this, and IMO, the "stretch" shouldn't be the central issue/

You do have to drop your wrist almost directly under the neck, so your pinky doesn't mute the 1st string. So, the first joint of the pinky should be very close to a right angle with the neck

There is more of an issue of strength for this fingering, than for stretch. If sufficient hand strength isn't present, the player tends to keep the wrist behind the neck, because it's easier to develop more leverage from there.