Ortho-para directing and meta directing groups: Activating or deactivating the aromatic ring?
I'm not sure if G is an activating or deactivating compound.

You may want to look at the relative order of activation/deactivation of some substituents,
(source: www.chem.ucalgary.ca)

As per the image, $\ce{-NR2}$ is a strongly activating group, while $\ce{-CONH2}$ is a moderately deactivating group (bcoz, it lies somewhere in between $\ce{-COOH}$ and $\ce{-COCl}$). Hence, the net effect would be activating.

How can resonance structures help to explain the reactivity of the compound?

Resonance structures can tell whether the substituent is o,p -directing or m -directing. It can't say anything about rate of reaction.

How do we draw resonance structures that has two substituents such as G?

Just like you've drawn for a single substituent. The only difference is that the other substituent has replaced one of the $\ce{H}$ from the resonance structure drawn with single one. So, the other may participate in delocalization.

For F, does the lone pair on N affect the resonance?

Obviously, but not significantly, as it will participate in resonance too.

I think Br will attach para to N because N is a strong activating group and it will not go to between the two substituents because it is crowded, is this true?

Yes, but it is partially true.
para product (wrt. $\ce{-NR2}$) will be major, however, ortho product (wrt. $\ce{-NR2}$) will be also present as the minor one.