Prove that $X^\ast$ separable implies $X$ separable
To cut a long story short: I don't really follow what you're doing in “(i) (by contradiction)” because the notation is quite confusing, but the argument for “(ii) Using:...” is basically okay, even if only barely readable.
A few remarks:
You don't say what $\{x_{n}^\ast\}_{n=1}^\infty$ is. Clearly, there is the implicit assumption that $D = \{x_{n}^\ast\}_{n=1}^\infty$ is a countable norm-dense subset of the Banach space dual $X^\ast$ of $X$.
Always state what you assume! For instance, your choice of $Y$ later on indicates that you work with a real Banach space (who should know that this is assumed if you don't state it?)
Fix $n$. By definition of the norm on $X^\ast$ we have $$ \|x_{n}^\ast\|_{X^\ast} = \sup_{\Vert x\Vert=1} |x_{n}^\ast (x)|. $$ The definition of the supremum provides us with an $x$ such that $\|x\| = 1$ and $\|x_{n}^\ast\|_{X^\ast} \geq |x_{n}^\ast(x)| \geq \|x_{n}^\ast\|_{X^\ast} / 2$ (if $x_{n}^\ast = 0$ take any $x$ with $\|x\| = 1$).
Multiplying $x$ with an appropriate scalar $\lambda = e^{i\alpha}$ (or $\pm 1$ if indeed we work over the reals), we can put $x_n = \lambda x$ in order to get $x_{n}^\ast(x_n) \geq \|x_{n}^\ast\|_{X^\ast}/2$. Do this for every $n$ to get a set $S = \{x_n\}_{n=1}^{\infty} \subset X$ such that for each $n$ we have $\|x_{n}\|_{X} = 1$ and $x_{n}^\ast(x_n) \geq \|x_{n}^\ast\|_{X^\ast}/2$.
This takes care of your point (i).
Let $S =\{x_n\}_{n=1}^\infty \subset X$ be the set chosen in 2 and let $Y=\operatorname{span}_{\mathbb{Q}}\,{S} \subset X$ be the set of rational linear combinations of elements of $S$ (if we should happen to work over the conplex numbers take $\mathbb{Q}+i\mathbb{Q}$ instead of $\mathbb{Q}$). Then $Y$ is countable (say that!). We want to show that $Y$ is dense in $X$, so that $X$ is indeed separable.
Note that the closure $\overline{Y}$ of $Y$ is a linear subspace of $X$ (why is that?).
Assume towards a contradiction that $\overline{Y}$ is not all of $X$. By Hahn-Banach we can find a continuous linear functional $0 \neq x^\ast \in X^\ast$ such that $x^\ast|_\overline{Y} = 0$. By norm-density of $D$ in $X^\ast$, we can find an $x_{n}^\ast$ in $D$ such that $\Vert x^\ast - x_{n}^\ast \Vert_{X^\ast} \lt 1/4$ . Because $x_n$ is in $\overline{Y}$ we have $x^\ast(x_n) = 0$ and thus $$ \frac{1}{2} \leq |x_{n}^\ast(x_n)| = |x_{n}^\ast(x_n) - x^\ast(x_n)| \leq \Vert x_{n}^\ast- x^\ast\Vert_{X^\ast} \, \|x_{n}\| \lt \frac14, $$ which is a contradiction. Done.
Note. I'm not sure if that was a typo, but actually we don't need the extra argument in 2. where we show that we can find $x_n$ such that $x_{n}^\ast(x_n) \geq \|x_{n}^\ast\|/2$ (without modulus signs).
You start the proof of (ii) by saying “Then $\{x_n\}$ dense in $X$” — wait, what?! are you assuming what you want to prove? — Ah, no, okay, you mean $\{x_{n}^\ast\}$ dense in $X^\ast$. pheew. You then conclude the argument more or less the same way as I did but your exaggerated formalism makes this extremely hard to follow.
Finally, a few things I have told you over and over again:
Get rid of all the symbols: $\forall, \exists, \implies, \iff, \lnot$ when you're writing stuff.
They are unpleasant to look at and hard to decipher.
Write prose. Use formulas only when you absolutely need them.
Please make complete, grammatically correct sentences. There is not a single complete sentence in your whole solution!
The idea of point 1. is that it forces you to observe point 2. and if you observe point 2. you are forced to eliminate all the quantifiers, implication sings and other logical symbols from your writing.
Please do have a look at
J. Milne's page: Tips for authors.
D. Goss: Some Hints on Mathematical Style based on tips by Serre.
I’m going to comment only on what you wrote for (i):
Assume for all $x_n \in X$ with $\| x_n \|_X = 1$ that
$$x_n^\ast (x_n) \lt \frac{\| x_n^\ast \|}{2} \iff 2 x_n^\ast (x_n) \lt \sup_{x_n; \| x_n \|_X = 1 } \{| x_n^\ast(x_n)|\}$$
Claim: Then $\exists r \in \mathbb{R}: x_n^\ast(x_n) \lt r \lt \sup_{\dots}\{\dots \}$:
(i) $x_n^\ast (x_n) \neq 0$: because $\| x_n \| = 1$ and $x_n^\ast$ linear
(ii) if $x_n^\ast (x_n) \lt 0$ then $2 x_n^\ast (x_n) \lt x_n^\ast (x_n) \lt x_n^\ast(- x_n) \lt 2 x_n^\ast(-x_n) \lt \sup_{\dots} \{ \dots \}$
(iii) $x_n^\ast(x_n) \gt 0$ then $\forall x_n: x_n^\ast (x_n) \lt 2 x_n^\ast (x_n) \lt \sup$
$\implies $ the $\sup$ is not the l.u.b., contradiction, so the claim $x_n^\ast (x_n) \geq \frac{\| x_n^\ast \|}{2}$ is true.
The first problem, and it’s a major one, is that right off the bat you’re using $n$ for two completely different things. On the one hand it’s apparently supposed to be the index of a particular, fixed member of some countable norm-dense subset of $X^*$, though you never actually said that. On the other hand it’s a dummy index picking out members of $X$ of norm $1$; this would be a bad idea even if you weren’t already using $n$ for something else, since there’s no reason to suppose that there are only countably many elements of $X$ of norm $1$. You should have begun something like this:
Let $D=\{x_n^*:n\in\mathbb{N}\}$ be a countable norm-dense subset of $X^*$. Fix $x_n^*\in D$, and assume for each $x\in X$ with $\|x\|=1$ that $$x_n^*(x)<\frac{\|x_n^*\|}2.$$
Note that I stopped before your $\iff$ symbol: that’s because what follows it is not part of your assumption, but rather an inference from your assumption, so it does not belong in the same clause with assume that. Once you’ve clearly stated your assumption, then you can go on and draw conclusions:
This implies that $$2x_n^*(x)<\sup_{\|y\;\|=1}\|x_n^*(y)\|$$ for each $x\in X$ with $\|x\|=1$.
Note that I had to use a different letter for the dummy variable ($y$) in the supremum from the one used for the specific $x$ of norm $1$ in the surrounding statement: they refer to different objects.
The line that begins Claim makes no sense even after the ellipsis at the end is properly filled in. (Note, by the way, that this is something that you should have done yourself, so that the reader needn’t guess; with cut-and-paste it’s completely trivial.) You still have $n$ meaning two different things, and it’s not clear whether you’re talking talking about a specific $x$ of norm $1$ or all of them together. I suspect that you meant this:
Claim: There is an $r\in\mathbb{R}$ such that $$x_n^*(x)<r<\sup_{\|y\;\|=1}|x_n^*(y)|$$ for each $x\in X$ of norm $1$.
Presumably what follows is supposed to be a proof of the claim. Say so.
Proof of Claim: Fix $x\in X$ of norm $1$. Since $\|x_n^*\|$ is linear, $x_n^*(x)\ne 0$.
This doesn’t appear to follow. In fact, there seems to be nothing in your argument to here to preclude the possibility that $x_n^*$ is the zero functional.
If $x_n^*(x)>0$, then $$x_n^*(x)<2x_n^*(x)<\sup_{\|y\;\|=1}|x_n^*(y)|\;,$$ and if $x_n^*(x)<0$, then $$2x_n^*(x)<x_n^*(x)<x_n^*(-x)<2x_n^*(-x)<\sup_{\|y\;\|=1}|x_n^*(y)|\;.$$
This does not in fact prove the claim; you’ve merely shown that for each $x\in X$ of norm $1$, $$x_n^*(x)<\|x_n^*\|=\sup_{\|y\;\|=1}|x_n^*(y)|\;,$$ i.e., that $x_n^*$ does not attain its norm on $\{y\in X:\|y\|=1\}$.
Had you made the effort to write it intelligibly, thinking about what you were actually saying, you might have noticed that much of it makes no sense and that it does not in fact do what you wanted it to do. At the very least you would have made it possible for others to spot the problems and help with them much more easily.
Remember: A proof is just a particular kind of expository prose. It should consist of paragraphs of sentences. Yes, it will often contain special symbols, but DON’T use symbols just for the sake of using them. The object is to convince the reader that something is true, and you can’t do that if you can’t make yourself understood.