1. Is it true that $U_P(\mathbb A) \subseteq G(\mathbb A_F)^1$ for standard parabolic subgroups $P \subseteq G$?

Yes. Let $x \in U_P(\mathbb A)$ be written as $x = umk$ where $u \in U_P(\mathbb A), m \in M_P(\mathbb A)$ and $k \in K$. Clearly $u = x, m = 1$ and $k = 1$. Then $H_P(x) = H_{M_P}(m) = 0$ so $x \in G(\mathbb A)^1$.

2. On the other hand, the second question is not true. For a quick counterexample, take $G = SL(2), P$ to be the minimal parabolic subgroup of upper triangular matrices and $g = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Note that $G(\mathbb A) = G(\mathbb A)^1$ and $g \in U_P(\mathbb A)$ but $g \neq 1$.

1. Is it true that $U_P(\mathbb A) \subseteq G(\mathbb A_F)^1$ for standard parabolic subgroups $P \subseteq G$?

Yes. Let $x \in U_P(\mathbb A)$ be written as $x = umk$ where $u \in U_P(\mathbb A), m \in M_P(\mathbb A)$ and $k \in K$. Clearly $u = x, m = 1$ and $k = 1$. Then $H_P(x) = H_{M_P}(m) = 0$ so $x \in G(\mathbb A)^1$.

Edit: If $H_P(g) = 0$ then $H_G(g) = 0$.

Observe that if $P_1 \subseteq P_2$ then $M_{P_1} \subseteq M_{P_2}$ and the restriction homomorphism $X(M_{P_2}) \to X(M_{P_1})$ is injective [Clay notes $\S 5$]. Thus $H_{P_2}(g) = 0$, which is equivalent to $\chi(g) = 0 \ \forall \chi \in X(M_{P_2})$ is true whenever $\chi(g) = 0 \ \forall \chi \in X(M_{P_1})$, which is equivalent to $H_{P_1}(g) = 0$.

2. On the other hand, the second question is not true. For a quick counterexample, take $G = SL(2), P$ to be the minimal parabolic subgroup of upper triangular matrices and $g = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Note that $G(\mathbb A) = G(\mathbb A)^1$ and $g \in U_P(\mathbb A)$ but $g \neq 1$.