I shall assume you are working on FA 2.4RC1 with all fixes till date.

Correct

Maybe I am not looking in the right tables, but nothing about the two accounts that create the counterparty column stands out to me.

I have already investigate this.  if This simply will not work out for me, I will find another machine to host FA until it can be integrated into the existing server.  I am keeping careful track of all my entries in case I need to redo them...

This is what I understand from investigating the definitions.  But does not seem to be consistent with how things are working in the interface...

enough for one day, will revisit again tomorrow...

As for the 4 GL Accounts issue, check their entries in the db tables for any record / field differences other than account code and name. Also check their entries in the sys_prefs table if the mischief is in it's being set as some default.

As you are using PHP7, there seems to be a sticky post in announcements that could be useful.

Counterparty is just the contra account in a transfer transaction. It can be between 2 bank accounts or even between some balance sheet entries and P&L entries like in depreciation entry transactions apart from the ones you already mentioned.

First; the best definition of counterparty I can find on the net is that this is the opposing party in the transaction.  In a supplier invoice, the counterparty would be my vendor.  in a customer invoice, the counterparty would be my customer.  I cannot find anything that makes me think this should be showing up in a journal entry anywhere.

I have 4 GL accounts configured as bank accounts, this only happens on one of them, my primary chequing account with GL account number 1060.  Here I get only one option in the drop down, and since that entry is both a customer and a vendor, I do not know where it is pulling it from, nor do I know why I am getting only one entry when it would seem ti should be all my vendors or all my customers, or both.

It also shows up on GL account 2100 Acounts payable, and there I get a full list of all my vendors.

I didn't test every single GL account, but I did test several accounts in each category, and these were the only two that showed the counterparty column.  I still dont' understand why only one option shows up under the 1060 account, nor do I understand how I am supposed to use this when I want to transfer money out of my chequing account, but not to a vendor and not to another bank account.  I am not sure this is working the way it is intended to?

Anyway, today I am starting to make entries (keeping track of course), and in making a journal entry - moving funds from chequing account to shareholders account - I am presented with something called "counterparty".  This is a drop down menu, it contains one item, and that item is an organization that happens to be both vendor and customer.  I don't know what counterparty is, and I have no idea why only this one option is listed.  Can someone enlighten me?