Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$ Yes, this kind of lower bound in $t$-aspect is trivial for any $L$-function. The point is that you can approximate $L(\frac 12+it)$ by some long Dirichlet polynomial $\sum_{n\le T^{r}} a(n)n^{-1/2-it}$, say, with $a(1)=1$, and the $L$-function coming from $GL(r)$, say. If you now integrate with smooth weights $L(1/2+it)$ (without absolute values), then note that only the term $n=1$ contributes. The rest of the terms cancel out and are negligible for smooth weights (rapid decay of Fourier transforms). So the bound $\gg T$ follows.

Is fourier analysis necessary to prove this? For part 2 note that the term $x=0$ is $1$, and the rest of the terms are at least $-2 \sum_{n=1}^{\infty} e^{-\pi n^2} = -0.086\ldots$.