When you get your tax stamp and all approved, Do you need to carry a copy of tax stamp with you anytime you are out in field hunting or at the range?

yes

I make a reduced size copy and stick it in the case with my hunting license

Absolutely you need to have a copy of the Tax stamp and your trust if applicable. I have the original in my safe and a colored copy that I had laminated in thick lamination. I keep the laminate copy with me in the field or at the range. I also made a 45% copy that is pocket sized and laminated it. I also keep a copy of my trust with me.

^^Do this^^

Thanks all for information

I have all the originals safely stored away, but I made a binder of all the copies and keep it with me. I used to keep each copy in the case that the weapon went into but it gets to be a pain. With the binder I have all the paper work with me if I'm just traveling with one or thirty NFA weapons.

When I copy a form, I go ahead and make an extra copy to leave in the safe. That way if something happens to the one in the binder I have a quick replacement or something to make a copy of without having to pull the original back out.

I keep a copy in the safe at the lease, in the truck, and in the safe at home. There is always one where ever I am.
It would be cool if the ATF would give you credit card size ID with the weapon(s) or suppressor(s) listed and drop all of the paper work (for $200 you would think they could afford this).
If you have a trust to get the stamp, it seems unnecessary that you have to keep a copy of the trust with you too.

If you have a trust with more than one member I'd think it'd be best to have it handy should someone in the trust besides the actual owner be checked. If I have the trust, and my cousin is a member and he is in his stand with my suppressor and gets checked he better have the trust with his name on it available. That's just how I see it.

This may be a STOOPID question, but does anyone file their federal tax stamps with their county clerk's office, like original birth/death certificates, DD-214's, etc?

edited to add: Of course, once filed with a county clerk it's public information provided someone knows what to ask for. Might be a bad idea.