Consider first the case when the $X_i$'s are independent. 
In view of your definition of a sub-gaussian random vector, we appear to have the condition
$$s^2:=\sup_i Var(X_i^2)<\infty.$$

Let $N:=\|X\|_2$. We have this key identity: 
$$N-\sqrt n=\frac{N^2-n}{2\sqrt n}-R_n,\tag{1}$$
where 
$$R_n:=\frac{(N^2-n)^2}{2\sqrt n(N+\sqrt n)^2}.$$
Moreover,
$$0\le R_n\le\frac{(N^2-n)^2}{n^{3/2}},$$
whence 
$$|ER_n|\le\frac{E(N^2-n)^2}{n^{3/2}}=\frac{Var(N^2)}{n^{3/2}}\le\frac{s^2}{n^{1/2}}\to0$$
(as $n\to\infty$). So, by (1),
$$EN-\sqrt n=-ER_n\to0,$$
so that your condition (1a) holds, which also obviously implies (1b). 

Your condition (2b) immediately follows from your second displayed inequality, $\|N-\sqrt n\|_{\psi_2}\le C$, which implies $\|\frac N{\sqrt n}-1\|_{\psi_2}\le C/n$. 

Your condition (2a) does not hold even when the $X_i$ are iid standard normal -- because then, by (1) and the central limit theorem (say), the distribution of $N-\sqrt n$ converges to $N(0,1/2)$. 

Thus, in the "independent" case, your conditions (1a), (1b), and (2b) hold, whereas (2a) does not hold in general. 

Consider now the "dependent" case, when the $X_i$'s are not necessarily independent. Let e.g. $X_i=X_1$ for all $i$, where $X_1$ is any zero-mean unit-variance random variable such that $a:=E|X_1|$ is strictly less than $1$. Then $N=\sqrt n\,|X_1|$. So, $\frac{EN}{\sqrt n}=a-1\not\to0$, so that (1b) fails to hold, and hence (1a) fails to hold. 
Also, here $\|\frac N{\sqrt n}-1\|_{\psi_2}=\||X_1|-1\|_{\psi_2}\not\to0$, so that (2b) fails to hold, and hence (2a) fails to hold. 

Thus, in the "dependent" case, none of your conditions (1a), (1b), (2a), and (2b) holds in general.