How do I find the transfer function in the frequency domain?
Your final answer looks good, but the derivation and notation are a bit mixed.  Remember the capitalized $H()$ refers to the frequency domain transfer function, so writing $H(n)$ with the time-domain index $n$ doesn't make sense.  Also, it is the time-domain impulse response that is a delta function, not the frequency domain transfer function, so $H(f)=-\delta(f)$ is not right.

The problem can be solved very easily:

$y(n)=-x(n)$

Take the Fourier Transform of both sides:

$Y(f)=-X(f)$

Divide both sides by $X(f)$:

$\frac{Y(f)}{X(f)} = H(f) = -1$

In the time domain:

$h(n)=-\delta(n)$