The answer to both questions is yes.

1. All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan, Representations of the group ${\rm GL}(n,K)$ where $K$ is a local field, Lie groups and their representations.

2. All ingredients to prove 2. are in

Paskunas, Vytautas; Stevens, Shaun On the realization of maximal simple types and epsilon factors of pairs. Amer. J. Math. 130 (2008), no. 5, 1211–1261.

First it is known by Bushnell and Kutzko that any irreducible supercuspidal representation of $G$ is of the form $\pi ={\rm ind}_J^G \Lambda$ (compactly induced representation), where $(J,\Lambda )$ is a maximal simple type in the sense of B. and K. Paskunas and Stevens prove that one can arrange the data so that $Hom_{U\cap J} (\Lambda , \chi ) \not= 0$ (1). It is a classical fact that if $c$ is a coefficient of $\Lambda$, then viewed as a fonction on $G$ (extend by $0$ off $J$), $c$ is a coefficient of $\pi$ (of the form $f_{v* ,v}$) (cf. e.g. Carayol Ann. Scient. ENS). Now writing condition (1) as the non-vanishing of an integral, you get exactly what you want.

Tell me if you some more detail.

The answer to both questions is yes.

1. All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan, Representations of the group ${\rm GL}(n,K)$ where $K$ is a local field, Lie groups and their representations.

2. All ingredients to prove 2. are in

Paskunas, Vytautas; Stevens, Shaun On the realization of maximal simple types and epsilon factors of pairs. Amer. J. Math. 130 (2008), no. 5, 1211–1261.

First it is known by Bushnell and Kutzko that any irreducible supercuspidal representation of $G$ is of the form $\pi ={\rm ind}_J^G \Lambda$ (compactly induced representation), where $(J,\Lambda )$ is a maximal simple type in the sense of B. and K. Paskunas and Stevens prove that one can arrange the data so that $Hom_{U\cap J} (\Lambda , \chi ) \not= 0$ (1). It is a classical fact that if $c$ is a coefficient of $\Lambda$, then viewed as a fonction on $G$ (extend by $0$ off $J$), $c$ is a coefficient of $\pi$ (of the form $f_{v* ,v}$) (cf. e.g. Carayol Ann. Scient. ENS). Now writing condition (1) as the non-vanishing of an integral, you get exactly what you want.

Tell me if you want some more detail.