Is there a systematic way to know when to alpha-transform free variables?
The answer here is the same as in the other question: one thing is missing here!

Your addition result should be:

$$3 + 4 = \lambda g . \lambda z . 3 g (4 g z) = \lambda g . \lambda z . 7 g z$$

Note that $g$ is now a lambda parameter, not a free variable! So now if you want to apply this to something, it'll get substituted in the same everywhere:

$$7 q r = (\lambda g . \lambda z . 7 g z) q r = q q q q q q q r$$