@IgorBelegradek, thanks for the comment. I understand this depends on the parametrization, but I exactly want to capture that (i.e. not allow for any parametrization), but stick to a given one that whomever gave me $w$ chose. When I say lift (technically) I mean a lift of curves from the sphere to the orthonormal frame bundle. I think my usage of the word 'lift' to denote taking a curve in Euclidean space to the sphere was technically incorrect.

Thanks for your answer. I hesitate to mark it as accepted because it doesn't really address Question 2, but if you think I should accept it anyway I will. Can you say more about the map $i$ which you refer to? What is a good reference for this?

@MarkGrant, in relation to my question here math.stackexchange.com/questions/2885139/… would you mind explaining why the suspension of the attachment map is nullhomotopic? The last answer to the linked question seems to have quite a big argument to support that; what is the way to argue here?

@AllenKnutson, I see, thank you. So we can think of X as the "Klein Bottle", and if $j=0$, the range of the maps is $S^4$, if $j=1$ the range is a closed 4-dim disc embedded in $\mathbb{R}^5$, if $j=4$ I think we get a 1-dim line in $\mathbb{R}^5$. Does that make sense? Not sure about the other cases. At any rate, how do you then go about classifying those maps under regular homotopy? How do you relate $\pi_2(Y)\equiv\pi_{S^2}(Y)$ for some $Y$ to $\pi_{Klein}(Y)$?

@WłodzimierzHolsztyński, yes, it is, I just wrote the parametrization explicitly to specify $f_1$. AllenKnutson, are you saying X/~ is the Klein-bottle? I don't see that.. I'm also unsure how you get from $G$-maps to quotient spaces.

Perhaps equivariant homotopy is unrelated to this problem. Is there a way to relax the continuity condition on the group action? But still define a $G$-map $f:X\to S^4$ to be a map such that $f(f_1(g, (z,\theta))) = f_2(g, f(z, \theta))$.

@AllenKnutson, thanks for your comment. It appears I made a mistake in the formulation of $f_1$ and so I have edited my question to fix it. I hope now it makes more sense that I have defined $f_1$ in three (or two if you will) lines. Since you have two circles at the boundaries of $X$, I wasn't sure if I could retract the cylinder into one circle or two, and if I retract it into two circles, what are the conditions between the two circles then, if any.