Question: What if I consider null set or singleton set as subsets of $\mathbb{R}$, surely they are connected and so are they considered as intervals?

Answer: 
In my opinion this is the best way to define an interval: $A\subseteq\mathbb{R}$ is an interval if $x\in A$ for all $\inf A<x<\sup A$, when of course the supremum and infimum might be infinite as well. It is easy to see that a singleton is an interval. (there are just no values of $x$ such that $\inf A<x<\sup A$ in that case, so it is trivially an interval). 

And yes, a subset of $\mathbb{R}$ is connected if and only if it is an interval.
$(a,a)=\emptyset\,,\forall a$ and $[a,a]=\{a\}$, so sure, why not?