I am wondering if there is a fast way that we can determine if a function is pointwisely convergent or uniformly convergent, without having to go through the formal proof?
It doesn't have to be a perfect property. It would be fine as long as it works well heuristically. For instance, a fast way I have so far discovered for determining whether a continuous function is uniformly continuous: if a function is continuous but unbounded (like $\frac{1}{x}$), then it is definitely NOT uniformly continuous.
Is there a similar rule for uniform convergence?
Thanks!
Broadly, the answer is no, because you can create a lot of tricky functions that converge in weird ways, and in fact I'm pretty sure it is possible to create a sequence whose convergence is unproveable (by pegging its convergence on another unproveable statement, e.g. something involving the sum of reciprocals of Godel numbers of true statements).
However, there are a bunch of results about convergence that you can use to make it easier to identify some convergent and divergent functions. You've already pointed out some of the tricks for divergence, but one particularly useful result for proving convergence is the Squeeze theorem, which states that (roughly) if the function you're looking at always sits between two other functions, and those functions converge to the same value at a particular point, then the function you're looking at also converges to that value.
For example, $-x \leq x \sin x \leq x$ for all $x$, so you can immediately state that $\lim_{x \rightarrow 0} (x \sin x) = 0$ since the other two functions also converge to $0$. There's an equivalent squeeze theorem for sequences as well, and it's often much easier to prove convergence through the squeeze theorem than via first principles.