First order axioms for sequence theory? In a set the repetition and order of the elements does not matter.
In a multiset the order fo the elements does not matter, but the repetitions do.
In a sequence both the order and the repetition matter.
Is there any first order logic and axiomatic formulation of sequences, more or less comparable to the ZFC axioms of set theory?
 A: I'm not sure that I correctly identified the point of your question. So please let me know whether the following answer is what you are looking for:
Within any reasonably rich set theory we can formalize the notion of multisets and sequences. $\operatorname{ZFC}$ is, obviously, an overkill to do so. But it's the theory most mathematicians (even if they don't know it explicitly) are familiar with. So, for the sake of this answer, let's work within $\operatorname{ZFC}$.
The "standard" approach to multisets and sequences, within $\operatorname{ZFC}$ is as follows:
Step 1. Unordered pairs.
Let $V$ be our set theoretic universe. We want to find a (definable class function)
$$
\langle, \rangle \colon V \times V \to V, \ (x,y) \mapsto \langle x, y \rangle
$$
that satisfies the fundamental property of a pairing function. Namely, for all $a,b,c,d \in V$,
$$\langle a,b \rangle = \langle c, d \rangle \iff a = c \wedge b = d$$
The Kuratowski pairing function
$$
\langle a, b \rangle := \{ \{a\}, \{a,b\} \}
$$
does provably satisfy this condition (in $\operatorname{ZFC}$).
Step 2. Cartesian products.
Once we have a pairing function, we can define the Cartesian product of two sets $X,Y$ as
$$
X \times Y := \{ \langle x,y \rangle \mid x \in X \wedge y \in Y \}.
$$
(This apparently circular use of Cartesian products can be avoided by a taking a little more care and talking about the actual formula defining the pairing function above.)
Step 3. Functions.
Once we have Cartesian products, we can define functions. A function $f$ with domain $X$ and range $Y$ is a subset
$$
f \subseteq X \times Y
$$
such that for all $x \in X$ there is a unique $y \in Y$ such that $\langle x, y \rangle \in f$.
Step 4. Multisets.
By the axiom of infinity, the set of natural numbers (including $0$!) $\mathbb N$ exists. A multiset $M$ with domain $X$ is a function
$$
M \colon X \to \mathbb N
$$
$x \in X$ is a member of $M$ iff $M(x) > 0$ and $M(x)$ is the multiplicity of $x$ in $M$.
It's easy to define the usual operations on multisets. For example, if $M, N$ are multisets with domain $X$ and $Y$ respectively, we can define the union multiset $M \sqcup N$ with domain $X \cup Y$ as follows:
$$
M \sqcup N \colon X \cup Y \to \mathbb N, \ z \mapsto M(z) + N(z).
$$
Step 5. Sequences.
A (countable) sequence $S$ with domain $X$ is a function
$$
S \colon \mathbb N \to X
$$
where $S(n)$ is the $n$-th element of the sequence $S$.
Viewed this way multisets and sequences are sort of dual notions and - modulo the intended interpretation - actually are the same objects when there domain happens to be $\mathbb N$.
