I noticed that when I view my list of highlights in Moon Reader, it shows the highlights from the very beginning of the book unless I mark "Sort by time", but then while it does show the latest highlight first, it goes in reverse order so that I see the latest highlight on top of the screen and than earlier highlights below. I would rather follow highlight from up to down rather than down to up.

Basically I want it to show the highlights in the order that it occurs in the book, not in the order that I highlighted it, but I also want it to show where I am in the book instead of going to the beginning of the book when I look at the list of highlights (like how it works on Kindle app on Android)

Is there no way to show the list of highlights as I want it show it?
you need to ask the author, though finding a current, working email for that may be hard.
By author do you mean the maker of the app?

Does the problem I mentioned happen in all Moon Reader or does the latest version of Moon Reader not have that problem? I tried it on Kitkat (on an old Android phone) and Nougat (on somewhat old tablet)

I also tried it on Pie, but don't remember if it had that problem or not, and can't check at the moment because my phone broke.
I meant ask the maker of the app, yes.
I don't use highlights much myself.
My current version is 6.20. we have been at version 6 for 2 or 3 months.

App release notes say pro users get email customer support, but I think there is an issue of what email address works
Could you ask the maker with the email customer support if you are using Pro version?

If I buy the pro version to email them and end up not getting a response or if they are unwilling to implement the function that I want, then I just wasted some money for app that I won't use.
YES

the cost of the app is about one starbucks coffee
take the risk, or move on
there is a list somewhere of the difference between Moon Reader, and Moon reader Pro. I don't recall seeing anything about highlights, so I would expect highlights to work identically in both versions.

if you don't mind a no longer supported or updated, and broken for Android 10 app, then try Bookari- it may do what you want.
I wanted to try to see if highlights synced by using a second Android, but I lost my Androi tablet at the moment and so I'm not able to try it out.

Is it certain that highlights don't sync? Only latest page syncs but not highlights?