Example where $(V^*)^*\neq V$?

Since you use the tag "functional analysis", I'll assume that you talk about continuous duals.

Let $V=c_0$, that is the space of complex sequences that converge to zero. It is well-known (and fairly easy) to show that $V^*=\ell^1(\mathbb N)$, and $V^{**}=\ell^\infty(\mathbb N)$.

In both cases the duality is $$\langle x,y\rangle=\sum_nx_ny_n.$$

The space $c_0$ is norm-separable, while $\ell^\infty(\mathbb N)$ isn't.

As for your second question, yes, you always have a canonical injection $V\to V^{**}$.

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For completion's sake, the answer is similar if we consider the purely algebraic dual. Take $V=\mathbb R[x]$, the space of real polynomials. It has a countable basis, namely $\{x^n\}_{n\in\mathbb N\cup\{0\}}$. For each $t\in\mathbb R$ consider $\phi_t:\mathbb R[x]\to\mathbb R$ given by $\phi_t(p)=p(t)$. The uncountable set $\{\phi_t\}$ is linearly independent, so $\dim V^*>\dim V$. Then $\dim V^{**}\geq\dim V^*>\dim V$ and they cannot be isomorphic (the fact that the dimension of the dual is at least that of the space follows from considering dual basis).

One can give a simple argument based on cardinality counting for the algebraic case.

Suppose that $V$ is a vector space of dimension $\mathrm{card}(I)$. Then $V$ is isomorphic to a direct sum $k^{\oplus{I}}$ of $I$-copies of $k$. Next

$$\left(k^{\oplus I}\right)^* = \mathrm{Hom}_k(k^{\oplus I},k) \cong k^{\prod I}$$

where the right hand side is the product of $I$ copies of $k$ (considered as a vector space over $k$). Suppose now that

$$\mathrm{card}(k) < \mathrm{card}(I)$$

Then

$$\mathrm{card}\left(k^{\prod I}\right) \geq \mathrm{card}\left(2^I\right) > \mathrm{card}(I) = \mathrm{card}\left(k^{\oplus I}\right)$$

This implies that under the assumption $\mathrm{card}(I) > \mathrm{card}(k)$ we have

$$\mathrm{card}(V^*) > \mathrm{card}(V)$$

In particular $\mathrm{dim}(V^*) > \mathrm{dim}(V) = \mathrm{card}(I)$. Now you can use exactly the same argument to show that

$$\mathrm{card}(V^{**}) > \mathrm{card}(V^*)$$

So in general $V$ and $V^{**}$ do not have the same cardinality.

For the first question. For every vector space $V$ it is true that $$ \dim(V)\leq \dim(V^*) $$ with equality if and only if $V$ is finite dimensional (see Theorem 3.12 in Advanced Linear Algebra by Steven Roman). Thus, if $V$ is infinite dimensional, then $V\not\cong V^{**}$, so every infinite dimensional vector space serves you as an example.

For question 2: Yes.