Take a look at Seated halves on Heritage Auction's website, and you will see that most coins with that much color have some variation of a toning pattern that consists of blue around the periphery, fading to rose in the center. It isn't impossible to find one with a toning pattern similar to yours, but it does not seem too common. It might just mean that yours has progressed further. However, PCGS may feel that the coin was stripped and re-toned. Or, they may have seen something that does not show in the pictures.
To answer your questions:
1) I would not send it to NCS, if it were my coin. There is no contamination to be removed, and the toning has not reached the terminal state, in my opinion. I would be afraid that removing the toning would lower the value at best, and possibly uncover some hairlines or other problem at worst.
2) If NGC sees a problem, they will also give it a "Details" grade.
3) In its current holder, it is probably a $900 coin, give or take. With a clear grade, it could jump to $1500 or better.
I just saw RWB's answer. He makes a good point about the lack of luster in the photos. If it looks dull in hand, you may be better off not spending more money on it.
Edit: Oops, sorry - forgot my manners. Welcome to the forum.