Assume that there is a continuous function $\hat{H}$ making the following diagram commutative:
$$\require{AMScd} \begin{CD} I \times I @>{\hat{H}}>> \mathbb{R}^2 - \{0\}\\ @VVV @VVV \\ I\times I @>{H}>> \mathbb{R}^2 \end{CD}$$
where the arrow below $\hat{H}$ should be a dotted arrow because we are searching for this function. And I am not skillful in drawing commutative diagrams this is why I draw $I \times I$ 2 times because I do not know how to draw one dotted arrow coming out of $I \times I$ going directly to $\mathbb{R} - \{0\}$ my bad.
I want to answer the following questions:
$(a)$ Show that there exists a homeomorphism $ \mathcal{l}: S^1 \rightarrow \partial(I \times I)$ such that the composition $\hat{h} \circ \mathcal{l}$ satisfies $$\hat{h} \circ \mathcal{l}(\mathbf{x}) = \mathbf{x} \textbf{ for } \mathbf{x} = (1,0),(0,1)(-1,0) \textbf{ and } (0, -1)$$
I got a hint to use the proof here in this link:
Sketching the image of the function $\hat{h}.$
Which is given below:
The key observation is that you know a little bit about what $\hat{h}$ looks like.
Assume $\alpha : (0,0)\to (1,1)$ and $\beta : (0,1)\to (1,0)$ are your paths.
Then $$\hat{h}(0,0) = \alpha(0)-\beta(0) = (0,0)-(0,1) = (0,-1).$$ as $t$ runs from $0$ to $1$, we get $$\hat{h}(t,0) = \alpha(t)-\beta(0) = (q,r) - (0,1) = (q,r-1).$$ We know almost nothing about this path except that $0\le q\le 1$ and $-1\le r\le 0$ and $(q,r)\ne (0,1)$, so drawing any path that stays inside this square and doesn't touch the origin is acceptable.
Then $\hat{h}(1,0) = (1,1)-(0,1) = (1,0)$. For $0\le s\le 1$, $\hat{h}(1,s)$ stays in $[0,1]\times[0,1]\setminus\{(0,0)\}.$ Again, drawing any path that stays in here is acceptable. Next $\hat{h}(1,1) = (1,1)-(1,0) = (0,1)$, and as $t$ goes from $1$ back to $0$ on the next segment, $\hat{h}(t,1)$ stays inside $[-1,0]\times [0,1]$ and doesn't touch the origin.
Finally, $\hat{h}(0,1) = (-1,0)$, and as $s$ returns to $0$, $\hat{h}(0,s)$ stays inside $[-1,0]\times[-1,0]$ and doesn't touch the origin before returning to $(0,-1)$.
You'll notice that any path you try to draw that satisfies these requirements forms a counterclockwise loop about the origin that is nonzero in the fundamental group, contradicting the assumption of the existence of the extension $\hat{H}$.
Note that this proves the sets in your puzzle cannot exist if they are path connected.
And also to use the following homeomorphism:
$$ \ell \colon S^1 \to \partial(I \times I)\colon (x,y)\mapsto\begin{cases} (0,0) & x=y=0 \\\left(\frac{x}{|x|}\sqrt{x^2+y^2},\frac{y}{|x|}\sqrt{x^2+y^2}\right) & |x|\geq|y| \\ \left(\frac{x}{|y|}\sqrt{x^2+y^2},\frac{y}{|y|}\sqrt{x^2+y^2}\right) & |x|<|y|\end{cases}$$
But when I tried to compose $\hat{h}$ in the given proof, the composition satisfies what we want for the point $(1,0)$ but it did not work for the point $(1,0)$ and also I do not have a definition for $\hat{h}$ in the proof in case of the points $(-1,0)$ and $(0,-1).$
Could anyone help me in adjusting the solution for this problem?
Note: This question is an extension to the question here (I got the hint in this link) Showing the existence of a homeomorphism $ \mathcal{l}: S^1 \rightarrow \partial(I \times I). $
This is too long for a comment, but it's not an answer.
Your question needs careful editing and careful thinking, because it's kind of a mess right now. Normally I'd sit down and write an answer, but to be honest, I have no idea what it's asking...which is a bad sign, because I've been doing stuff like this for over 40 years.
Let's start at the top. We don't need to assume there's such a function $\hat{H}$ -- we can just write one, along with $H$, namely
$H(s, t) = (s + 3, t + 3)$
Define $H$ to be essentially the same function, with the codomain restricted. Done.
Part (a) of your question asks about a function $\ell$ related to some other function $\hat{h}$. I'm guessing you meant to write $\hat{H}$, but you wrote it wrong multiple times, so I have no idea. Probably you just mislabeled the diagram at the top.
AFTER part a, you seem to be telling me something about the function $\hat{h}$. That's like saying "Give me a number between 24 and 26" ...."oh, and it needs to be prime."
Seriously -- take some pride and formulate your question in a way that goes from stuff you know ("I am trying to show that for a function $h$ with the following properties,...") to stuff you don't know, as in "... there's no function $\hat{h}$ satisfying ..."
Please don't make me go off and read an equally-badly written related question -- just state your hypotheses and desired conclusion, and let us try to help out with a proof or counterexample. It's fine to mention the related question, but your question should be self-contained.
If you'd like to get answers useful to you, you might give hints about your background ("I don't know any homology or homotopy theory --- just basic 2nd-year calculus stuff, and a definition of winding-number using integrals...") And after you've stated the problem really clearly, THEN tell us what you've tried, and where you got stuck, so we don't feel like we're doing your homework for you.
Do all that, and maybe someone will get energetic and provide an answer.
(You might also want to look at How to ask a good question -- it's a pretty good summary.)