What should I do regarding my passport and visa extension in Thailand?

Get the new passport first or you just waste time to get a 5 month extension, and have to go thru the entire process again in July.

Should I get a new passport now or after I've done my extension?

-> Generally, getting a new passport would make many things easier. Still, it's totally your decision.

Is there some kind of rule that there must be at least 6 months of validity of my passport for a new extension?

-> No. But your extension will be until passport validity only. In your case, only until July 2024.

Will I be able to transfer the extension to the new passport and continue for the full validity or do I have to apply for a new extension again?

-> There is no "compensate" for that. You will apply for a whole new 1-year extension. So the timing of the yearly extension application will be changed to July after.

I will travel abroad in January 2024 and will obtain a re-entry permit. Does my passport need to have at least 6 months of validity to get the re-rentry permit and return to Thailand a week later?

-> No problem as long as you have a valid re-entry permit (means with a valid extension) and will be back within the validity. Thailand doesn't have 6-month passport validity rule anyway. You better check with your airline.