1) Yes, it is true that any closed submanifold $X_n \subset \mathbb R^{n+1}$ is orientable , even if $X$ is not compact.
Once you have orientability, the normal bundle is necessarily trivial.
Indeed, there exists an oriented frame $X_1, X_2,...,X_n\; (X_i\in \Gamma (X,TX))$ for $X$.
For every $x\in X$ there are two vectors in $ T_x(\mathbb R^{n+1})$ orthogonal to $T_xX$ and of length $1$.
By selecting $n(x)$, the one such that the basis $X_1(x), X_2(x),...,X_n(x),n(x)$ of $T_x(\mathbb R^{n+1})$ is direct, you obtain a nowhere zero section $n\in \Gamma (X,N)$ trivializing $N$.
2) No, this is false: some manifolds have no embeddings with trivial normal bundle in any $\mathbb R^{n+k}$. Here is why:
From the exact sequence of vector bundles on $X$
$ 0\to TX\to T\mathbb R^{n+k}|X \to N\to 0 $ you deduce the equality of Stiefel-Whitney classes $w_1(TX)=w_1(N)$.
So if the normal bundle $N$ were trivial, you would conclude that $w_1(TX)=0$.
But this is false for all even dimensional real projective spaces $\mathbb P^{2r}(\mathbb R) $ .
So in any embedding $\mathbb P^{2r}(\mathbb R)\hookrightarrow \mathbb R^{n+k}$ these projective spaces have non-trivial normal bundles.