No I mean, for bijective constructions between countably infinite sets and between finite sets, we can easily write down the resulting sequence to work out how to construct the bijective function

I know that I can verify that some real function is bijective. What I am trying to learn is how to find them no matter how crazy the uncountable set is. Unlike countable sets, the reals cannot be wrote into a sequence, and if you don't know the function beforehand, the only help from the diagram is you know what the required image of that function should be.

What type of object is the infinity we use in analysis, pretty much the only thing I knew is that it is defined to be larger than all finite numbers ($\aleph_0$ also have the same property), Does it have ties with set theory at all?

We can have uncountable index set. Thus the representation may be written as $\sum_{i \in I}b^ix_i$ where $b$ is the base. Granted, such expansion cannot be even written down as some sequence, but I don't see anything inconsistent here nor it is explicitly ruled out (and we can then have real number bases....? )

"Proof" that $|\Bbb R| = |\Bbb N|$:
1. It is a fact that $|\Bbb Q| = |\Bbb N|$.
2. In the Dedekind cut definition, a real number is defined as a partition of $|\Bbb Q|$ where the larger set is not empty.
3. For example, if you have 5 objects in a row, you would have exactly 5 ways to partition them such that the right set is not empty.
4. By analogy, since $\Bbb Q$ is ordered, consider it as a row, and the number of ways to partition it would be $|\Bbb Q$.
5. Since each partition defines a real number, the result follows.