Can we have an uncountable number of isolated points? Is this possible? I've been trying to think of an example or defend why not, and I'm struggling in both directions.
Since you use the real-analysis tag, I will assume that you’re talking about $\Bbb R$ with the usual topology. If so, the answer is no: every discrete subset of $\Bbb R$ is countable. 

To see this, note first that the set $\mathscr{B}$ of open intervals with rational endpoints is a countable base for the topology. Now suppose that $D\subseteq\Bbb R$ is discrete. Then for each $x\in D$ there is a $B_x\in\mathscr{B}$ such that $B_x\cap D=\{x\}$. $\mathscr{B}$ is countable, so if $D$ were uncountable, there would have to be distinct points $x,y\in D$ such that $B_x=B_y$, which is absurd: that would mean that 

$$\{x\}=B_x\cap D=B_y\cap D=\{y\}\;,$$

yet $x\ne y$.

If you are asking the question in general, however, rather than about $\Bbb R$ with the usual topology, then the answer is yes; Ross Millikan has given a simple example.
No, each neighborhood around each point is disjoint and contains a rational not contained in any other point's neighborhood.