Chemistry Question

I feel so weird asking a question about subjects in school because I am usually one of the ones who completely and totally understand concepts, but I am having a HARD time wrapping my head around a few things so any help will be greatly appreciated.

I have great trouble determing how many lewis structure bonds are possessed by a central atom. For example in the nitrate ion. Our instructor gave us this formula for determing bond number

total electrons-(electron pairs x 2)/2=total bonds.

So, N has 5 electrons, O has 6 each (18 for O) and one for the anion componant, for a total of 24 electrons.
Nitrogen has one pair, and each oxygen has 2 pairs, so 1+2+2+2=7 pairs
7 x 2= 14 paired electrons and 24 total electrons-14 paired electrons=10/2=5 bonds----and there is only 4 bonds in the nitrate ion. I am so totally confused.

But this formula WORKS for CH2O where 4+2+6=12 and oxygen is the only atom with paired electrons a total of 2. 2 x 2=4, 12-4=8/2=4 bonds, and CH2O has 4 bonds. Works, but doen't work for ions, does anyone know why? Or what formula I can use for ions---what am I doing wrong?

Also, I can't seem to wrap my head around any element in group 6A periods one and two can have more than 2 bonds with other atoms. There are only 8 electrons in their valence shell, so how in the world can Sulfur bond to six flourines when Sulfur already has 6 electrons and can only take on two more? Can anyone explain this to me? If it had d orbitals it would make sense, but as it is doesn't make any sense to me.

I have emailed the instructor and am waiting on a reply.

Any help is greatly appreciated!!