translation invariant of integral on $\mathbb{R}$

In your solution, it's not clear to me why you can just assert that $\int_{-n}^nf(x+t)\;dt=\int_{-n+x}^{n+x}f(t)\;dt$. This seems to be essentially equivalent to what you're trying to prove.

Instead, I'd proceed in the following way: first prove that $\int_{\mathbb{R}}f(x+t)\;dt=\int_{\mathbb{R}}f(t)\;dt$ when $f=\chi_E$ is a characteristic function (this should be immediate from the translation invariance of Lebesgue measure). Then prove it for simple functions by the linearity of the integral, then for non-negative functions by approximating with simple functions and using the monotone convergence theorem, and finally for arbitrary $L^1$ functions by linearity again.

For part 2, I suggest using the fact that the set of continuous functions with compact support is dense in $L^1(\mathbb{R})$, together with the fact that a continuous compactly supported function is uniformly continuous.