My Mac keeps asking me "do you want the application "SyncAppServer" to accept incoming network connections?" whatever I do

First, restart your Mac in "Safe Mode": Use safe mode on your Mac - Apple Support.

While in that mode, open Security & Privacy again and select the Privacy tab. In the column to the left, select each of the categories that appear, and determine if "SyncAppServer" appears in the column to the right. You will need to inspect each category separately. It is more likely to appear in Automation or Full Disk Access or Files and Folders than Contacts (for example). If you find it in the column to the right, select it and click the [—] (minus) button to prevent it from doing that.

For reference: Change Privacy preferences on Mac - Apple Support

You might not find it. Whether you find it or not, restart your Mac normally, and reply with your results.

If that does not help please download and run EtreCheck. A description and instructions are here: How to use the Add Text Feature When Posting Large Amounts of Text, i.e. an Etrecheck Report - Apple Community.

Post its report in a reply to this Discussion.

There are a tremendous number of system modifications affecting that Mac. Too many to enumerate. Any number of them can be responsible for the "SyncAppServer" nag.

I can only provide the following general advice.

System extensions installed - This computer has system extensions installed. System extensions can be difficult to uninstall.

That fact cannot be overstated. Anything you install on a Mac that requires authentication (Admin name and password) requires introspection, and nothing gets installed on a Mac gets installed without deliberately providing those credentials. Of course a Mac would not be the useful appliance that it is without that ability, but whatever you choose to install becomes something you need to diligently research and constantly maintain. In my opinion that Mac has been altered to such an extent that it is practically unmaintainable.

To reiterate, it's your Mac, and what you choose to install is up to you.

Now for some useful advice. No Mac requires third party "anti-virus" products, and installing them will only result in it not working as designed. In keeping with that principle uninstall "AdGuard" and "Sophos". Follow their respective uninstallation instructions, assuming you can find them, and that they actually work (which tends to be rare). But those are just two examples of a plethora of "stuff" affecting that Mac that no one needs.

That general advice should apply to any file appearing in any of the following three folders:

~/Library/LaunchAgents

/Library/LaunchDaemons

/Library/LaunchAgents

There should only be few files, if any, in each of the above folders. It is incumbent upon any Mac owner to have at least a passing familiarity with their contents. For example, Adobe, Google, Microsoft are well-known legitimate products, but even they need to be maintained and you need to be aware of the potential for them to cause adverse effects. All three of them have been known to do that.

That's only three... you have dozens.

Sure, that's perfectly ok. I think you fixed it.