For all integers $n$: if $7n+4$ is even, then $5n+6$ is even.

What I see is if $7n+4$ is even then $7n$ is also even, and so $n$ is even too.

Then $5n$ is even and thus $5n+6$ is even.

Walking through your attempt:

Claim: For all integers $n$: if $7n+4$ is even, then $5n+6$ is even.

Proof: Assume $5n+6$ is odd.

Normally proving $P_a\Rightarrow P_b$ you would start by assuming $P_a$. Here you're assuming $\lnot P_b$, suggesting you want to prove the contrapositive, $\ \lnot P_b\Rightarrow \lnot P_a,$ which is equivalent to $P_a\Rightarrow P_b$. But you need to understand where you are trying to get to.

If $5n+6$ is odd, then $5n+6=2k+7 \ldots$

This is unusual - it would be more usual to say $5n+6=2k+1 $

$\Rightarrow 5n=2k+1$ for some $n,k \in \mathbb{Z}$.

Which would then be $\Rightarrow 5n=2k-5$.

If $7n+4$ is even,

Having started in on the contrapositive, you are now switched over... perhaps for the purpose of contradiction, but it is starting to get hard to follow.

then $7n+4=2k$ for some $k \in \mathbb{Z}$.

You shouldn't reuse $k$. We'll say $7n+4=2m$ for some $m\in \Bbb Z$. This was your main mistake.

Therefore, $5n=7n+4+1$

Not true. This is the result of re-using $k$ inappropriately.

$\Rightarrow 2n=-5$ for some $n \in \mathbb{Z}$. Clearly, $2n \neq -5$, which contradicts our assumption that $5n+6$ is odd.

Actually - if this were valid - you should have it contradict your later statement that $7n+4$ is even, putting you on track to prove the contrapositive.

You don't need to come into the proof backwards on this occasion. It's easy enough to start with the the premise:

Taking $\mathit{ 7n+4}$ as even, we have $\mathit{ 7n+4=2m}$ for some $\mathit{ m\in \Bbb Z}$. Then $\mathit{ 5n+6 = (7n-2n+4+2) = 2m-2n+2 = 2(m-n+1)}$ and since $\mathit{ (m-n+1)\in \Bbb Z}$ we have ${\mathit 5n+6}$ is even.

Here is another approach. To make $5n+6$ appear in $7n+4$, consider their difference: $$ (7n+4)-(5n+6)=2n-2=2(n-1) $$ Therefore, $$ 5n+6=(7n+4)-(2n-2) $$ is even, being the difference of two even numbers.