What is the motivation for sequences to be defined on natural numbers?

Your main question seems to be

Why not define sequences on some other index set?

Okay, the term sequence is reserved for functions $a : \mathbb N \to X$ (note that I wrote $X$ instead of $\mathbb R$ because we may consider sequences in arbitrary sets, or if we are interested in convergence, in arbitrary topological spaces). However, this can easily be generalized to other index sets. This results in the concept of a net. See here. Quote:

Let $A$ be a directed set with preorder relation $\ge$ and $X$ be a topological space. A function $f: A \to X$ is said to be a net.

Recall that a preorder relation $\ge$ on a nonempty set $A$ is a reflexive and transitive binary relation. We do not require it to be antisymmetric (if it is, it is called a partial order). $(A,\ge)$ is called directed if for all $a, b \in A$ there exists $c \in A$ with $c \ge a$ and $c \ge b$. Simple examples are $\mathbb N, \mathbb Z, \mathbb Q, \mathbb R$ with their natural partial order. These sets are even totally ordered which means that for any two elements $a, b$ we have $a \ge b$ or $b \ge a$.

We often write nets in the form $(x_\alpha) = (x_\alpha)_{\alpha \in A}$. A net $(x_\alpha)$ is said to convergence to $x \in X$ if each open neigborhood $U$ of $x$ in $X$ admits $\alpha_0 \in A$ such that $x_\alpha \in U$ for all $\alpha \ge \alpha_0$. This obviously extends the concept of convergence from sequences to more general objects.

Interesting examples of nets which are no sequences occur in calculus when the Riemann integral is introduced. Here we consider partitions $\mathfrak{P} = (t_0, \ldots,t_n)$ of an interval $[a,b]$ and order them by"set inclusion" (i.e. $\mathfrak{P}' \ge \mathfrak{P}$ if $\{ t'_0, \ldots,t'_{n'} \} \supset \{ t_0, \ldots,t_n \}$). The uncountable set $P$ of all partitions is a directed partially ordered set, but it is not totally ordered. For any bounded function $f : [a,b] \to \mathbb R$ we consider the lower and upper Riemann sums which gives us two nets indexed by $P$. Both nets converge, but in general do not have the same limit. If they have, then $f$ is called Riemann integrable and the common limit is called the Riemann integral of $f$.

However, if you study calculus textbooks, you will find that frequently one again works with suitable sequences of partitions, e.g. equidistant partitions with mesh $(b-a)/n$. But note that conceptually one has to go beyond sequences.

An equivalent approach to the Riemann integral is by taking the elements of a modified index set $\mathbf P$ to be all systems $(\mathfrak{P},\mathbf x)$ where $\mathfrak{P} = (t_0, \ldots,t_n)$ is a partition of $[a,b]$ and $\mathbf x = (x_1,\ldots,x_n)$ with $x_i \in [t_{i-1},t_i]$. Defining $(\mathfrak{P}',\mathbf x') \ge (\mathfrak{P},\mathbf x)$ iff $\mathfrak{P}' \ge \mathfrak{P}$ we get a preorder on $\mathbf P$ which is directed but not antisymmetric. For any function $f : [a,b] \to \mathbb R$ we consider the Riemann sums $R(f;\mathfrak{P},\mathbf x) = \sum_{i=1}^n f(x_i)(t_i - t_{i-1})$. This system is a net over $\mathbf P$, and it is well-known that this net converges iff $f$ is Riemann integrable (and in that case the limit of the net is the Riemann integral of $f$). This shows once again that nets which look at first glance "fairly exotic" occur quite naturally in calculus and that we do need a concept of convergence for such nets.

So what is the reason for the prominent role of sequences in calculus? First of all, they are much simpler than general nets. For example, you can use induction. But the deeper reason is that all topological properties of $\mathbb R$ can be expressed via sequences. For example, $M \subset \mathbb R$ is closed iff for all sequences $(x_n)$ in $M$ which converge to some $x \in \mathbb R$ one has $x \in M$. Moreover, $M \subset \mathbb R$ is compact iff each sequence $(x_n)$ in $M$ has a convergent subsequence. Also the continuity of functions can be expressed via sequences: A function $f : \mathbb R \to \mathbb R$ is continous at $x$ iff for all sequences $(x_n)$ such that $x_n \to x$ one has $f(x_n) \to f(x)$.

Let me finally remark that series $\sum_{n=1}^\infty a_n$ should (at least in my opinion) be treated as nets. The usual approach is to consider the sequence $(s_m)$ of partial sums $s_m = \sum_{n=1}^m a_n$ and study its convergence. I suggest to introduce the set $\mathbf F$ of all finite subsets of $\mathbb N$ and order it by inclusion ($F' \ge F$ if $F' \supset F$). This produces a directed partially ordered set $\mathbf F$. Define a net over $\mathbf F$ by $(\sum_{n\in F}a_n)_{F \in \mathbf F}$. This net may converge or not. If it converges, we denote its limit by $\sum_{n\in \mathbb N} a_n$. It is now a nice exercise to show that $\sum_{n\in \mathbb N} a_n$ exists iff $\sum_{n=1}^\infty a_n$ is unconditionally convergent. Perhaps this unusual approach seems to be too exotic, but it has the benefit that it can be generalized to sums over uncountable index sets. Such sums occur in the context of Hilbert spaces. It is well-known that each Hilbert space $H$ has a (possibly uncoutable) orthonormal basis $\{b_\alpha\}_{\alpha \in A}$ and that each $x \in H$ can be written as $x = \sum_{\alpha \in A}\langle x, b_\alpha \rangle b_\alpha$.

When you have a sequence of real numbers, there is a first number in this sequence. Give it the index 1. Then there is a second number, give it the index 2. Continue. You might now ask: what is the real number with index $n$ in my sequence? For this, you define the function $f:\mathbb{N}\rightarrow\mathbb{R}$ which maps indices to the corresponding real numbers in your sequence.

When you now think a bit about it, your sequence is completely defined by the function $f$, and on the other hand $f$ is completely defined by the sequence. In other words, there is a bijection of sequences of real numbers and functions $\mathbb{N}\rightarrow\mathbb{R}$. This is the reason why we say that $f$ itself is the sequence.

Depends what you want/need. In applications as far as most of us are aware at least, the index will come from a countable set as you say.

For example thinking of annuities (amounts of money payed at regular intervals), we usually think of these as payments at discrete times for obvious reasons (possibly infinite) and so we would be treating them as a sequence of payments indexed by naturals (can apply to it the ‘usual‘ sequence theory).

If the payments are approximated/thought of as ‘continuous’ (indexed by $\mathbb{R}$ if you wish), then we use calculus (integration).