The stuttering sequences Let's define a stuttering sequence the following way :
Let $q\in\mathbb{N}^*,E_q=\{1,2,\dots,q\}$ and $(u_n)\in (E_q)^\mathbb{N}$.
$(u_n)$ is a stuttering sequence of order $k$ with spacing $w$ iff $$\exists n,w\in\mathbb{N},\exists k \in \mathbb{N}^*,\forall i\in\{0,1,\dots,k-1\},u_{n+i}=u_{n+w+k+i}$$
(Notations : $\mathbb{N}=\{0,1,2,...\},\mathbb{N}^*=\{1,2,...\}$)
Said a simpler way, $(u_n)$ is a stuttering sequence of order $k$ with spacing $w$ iff  it has a pattern of length $k$ that is repeated twice with a space of $w$ between the two patterns.
Examples :
Order $k=3$, spacing $w=0$ : $\{u_n\}_\mathbb{N}=\{\dots,9,1,2,\color{red}{1,5,9},\color{blue}{1,5,9},5,7,8,9,6,4,,\dots\}$
Order $k=4$, spacing $w=2$ : $\{u_n\}_\mathbb{N}=\{\dots,\color{red}{1,5,9,12},3,7,\color{blue}{1,5,9,12},\dots\}$
Remark : what is between the two patterns does not matter : $\{u_n\}_\mathbb{N}=\{\dots,\color{red}{1,5,9},\color{green}{1,5,9},\color{blue}{1,5,9},\dots\}$ is a stuttering sequence of order $k=3$, spacing  $w=0$ and spacing $w=3$.
My question :
Let $q\in\mathbb{N}^*$ :


*

*$\forall (k,w)\in\mathbb{N}^*\times\mathbb{N}$, is any sequence $(u_n)\in(E_q)^\mathbb{N}$ stuttering of order $k$ with spacing $w$ ?

*Can we find $(k,w)\in\mathbb{N}^*\times\mathbb{N}$ such that any sequence $(u_n)\in(E_q)^\mathbb{N}$ is stuttering of order $k$ with spacing $w$ ?

*For every sequence $(u_n)\in(E_q)^\mathbb{N}$, can we find $(k,w)\in\mathbb{N}^*\times\mathbb{N}$ such that it is stuttering of order $k$ with spacing $w$ ?

*(Added with an edit) The most interesting question : For every sequence $(u_n)\in(E_q)^\mathbb{N}$, can we find $k\in\mathbb{N}^*$ such that $(u_n)$ is stuttering of order $k$ with spacing $w=0$ ?
The cases $q=1,q=2$ are rather straightforward, but I haven't found any way to solve it for higher $q$s.

Update : User Carry On Smiling has answered points one and three, but points two and four are still open.
 A: Q4
No. See square-free words, they are sequences that are (in language of your problem) not stuttering for any $k \in \mathbb{N}^*, w=0$ and they are proven to exist for any $q>=3$.
Q2
No. Lets prove this for $q>=4$ by contradiction:
Let us assume that for some $q>=4$ holds: there are $k\in\mathbb{N}^*, w\in\mathbb{N}$ such that any sequence $(u_n)$ as described is stuttering with parameters $k,w$.
If $w=0$ then let $(u_n)\in(E_3)^\mathbb{N}$ be a square-free word as described in link above. It gives a required contradiction as stutter would produce a square in $(u_n)$.
Let us continue with $w>0$. We can construct a different sequence:
$$ (u_n) = (1^k2^w3^k4^w1^k2^w3^k4^w \dots) = (\underbrace{11 \dots 1}_{k\ elements} \underbrace{22 \dots 2}_{w\ el-ts} \underbrace{33 \dots 3}_{k\ el-ts} \underbrace{44 \dots 4}_{w\ el-ts} \underbrace{11 \dots 1}_{k\ el-ts} \dots)$$
By definition, if there is a stutter in $(u_n)$, then there is an index $s$ such that $\forall i\in\{0\dots{k-1}\}: w_i=w_{i+k+w}$. But for our sequence $w_i=w_{i+k+w}$ never holds: moving by $k+w$ elements switches from $1$ to $3$, from $2$ to $4$, from $3$ to $1$ and from $4$ to $2$. Therefore, $(u_n)$ is not stuttering with parameters $k,w$.
Therefore, initial assumption must be false.
For $q \in \{1,2,3\}$ proofs are more elementary, I am leaving this for the interested reader :)
A: Question $1$: The answer is no since the sequence $21212121\dots$ is not $1$ stuttering with $k=1$.
Question $2$: The answer is no, given any $k$ we can find a sequence using only $1$ and $2$ that is not $k$ stuttering with spacing $0$.
Proof: Take the sequence $\underbrace{00\dots0}_\text{k zeroes}1$ (and it repeats like this forever).
Question $3$:
We first prove that all sequences are suttering with $k=1$ and $w$ as large as desired. This is trivial, one digit must appear infinite times, take two appearances of that digit that are far enough from each other and let $w$ be the number of digits between them.
Suppose you have a sequence consisting of the digits $1$ through $n$, you want to prove it is $k$-stuttering with distance $w$. process the sequence to create a new sequence, there are $n^k$  possible subsequences of length $k$, so order those sequences lexicographically and convert that sequence into a new sequence where the first term is the number for the sequence of the first $k$ numbers. We now want to prove that this sequence is  stuttering with order $1$  and spacing larger than $k$. This is possible by the previous paragraph.
So given a sequence and a fixed $k$ we can find infinite values of $w$ for which it is $k$-stuttering with spacing $w$.
