@Yemon Choi: thank you for the suggestion, I will try that. I see a private note on top of the question saying "This post was edited and submitted for review 13 days ago", so I assumed somebody would review this eventually.

To Alex M.: I originally posted a longer version of the question that had three subquestions Q1, Q2, Q3, but the original question was closed because "This question does not appear to be about research level mathematics within the scope defined in the help center." I was given the option of editing the question and then I resubmitted a shorter edited version for review (but still the same question really). I hope they will open it again, because I think otherwise nobody can add further answers. Notice that I added a comment at the end the question acknowledging that it was edited.

Thank you, Gerry, I changed this to "four times as large" now.

I like your generalization to ${\rm deg}(v)=d>4$, it looks plausible. The requirement of maximality of the graph can potentially also be relaxed.

I edited the post now to stay "If the minimal degree of 𝐺 is smaller than five,..." to avoid similar confusion in the future. Sorry about this mistake.

Thank you for your reply. Indeed, instead of writing "If the maximal degree of 𝐺 is smaller than four, ..." I should have written "If the minimal degree of $G$ is smaller than five, ..." These are not the same statements, as you point out correctly. However, the claim regarding the four-color theorem is valid. For example, if there is a degree 3 vertex in the minimal counterexample we can just remove that vertex and the resulting graph should still be a counterexample. For degrees $\leq 4$ these are the arguments by Kempe from the 19th century.