Thank you everyone for your replies, let me go through and answer a few things, and give some more information where possible.
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First, let me thank you for the detailed reply and direct answers to questions, it was rather helpful. I am pleased to report that the e-mail does contain a direct link to the support ticket page, so I can easily keep a check on it. The status is still marked as 'Open', I will note. Five weeks for an initial reply sounds nerve-wracking, I'm glad things have clearly gotten better, as I response within 2 days I think.
I'll be waiting to see what happens this next week, and keeping the ability to send a chat message in mind if it becomes necessary. Thanks again.
Regarding the suggestions that this may be caused by the ISP, as noted in my detailed post I contacted the ISP as per GoG's instructions. The ISP told me they had no means on their end to throttle or block any URL, and as such could not be responsible. While I admit this sounds rather weird, it's the only answer I can get on the matter. Furthermore, literally every single other website I attempt to access is acting as usual, and every other page I try to access on GoG works without any error. The only page with any issues is the 'Order History' page, which displays the 'GoGbear' image.
Regarding the 'GoGbear' and the error that is causing it, I've used 'inspect element' to open the console to see if I could get any clues, and consistently it will read out '500 Internal Server Error' when refreshing on the 'Order History' page. So yes it seems safe to say that this caused by a 500 error on their end.
As for what may have caused this, I'll lay out a more detailed description of what I did/what happened that led to this, at the end of this post, momentarily.
I have always used the same e-mail with this account, and always used the same account name. I have never changed either one. As you say, things were working until I asked them to fix an item, and then it broke. That was while dealing with 'Customer Support'. I have since been transferred over to 'Technical Support', so hopefully that will help.
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As I said in my OP, I was trying to be brief, but now I think it's important that I go into a bit more detail on what happened.
I had 3 items in my cart, and decided to check out. i always pay using paypal, so was redirected to them to issue a payment. Before I could finish this I was distracted for a few minutes. Because some time had passed, and I am habitually one to check and recheck that I am doing the right thing, I clicked 'cancel' on paypal, to return back to the Gog card. From there, I rather quickly checked the cart, then hit 'checkout' again, once more being redirected to paypal where I finished payment, and was redirected back to GoG.
Instead of landing on the 'order complete' page, where they show off all the new game entries that have been added to your Library, I instead found myself back in the cart, and every item in said cart now was greyed out and said 'owned' next to it.
From there I opened a new tab and went to 'Order History', which showed 2 separate entries for the games I had wanted to buy (that is, two orders with the same games, and a different order number for each), and a button to 'Complete this Order' as next to the price total it read '(NOT PAID YET)'. Seeing this I immediately saved screenshots of both orders and the cart, as this was not the first time I had encountered this error (more on that in a bit).
From there I watched and waited, and eventually rather than either order completing, both orders vanished from history. This is where I then decided to reach out to Customer Support, in case there might be any issues that could appear later on. I explained the situation completely, including that there had been 2 orders that had appeared. I set up the support ticket using the order number from the one my paypal receipt said I had paid for. Customer Support got back to me soon, explaining that they had checked their records and saw that the error was on their end, and had reinstated the order in Order History.
It was here, when I attempted to load the 'Order History' page, that I was met with the 'GoGbear' rather than order history, and have been dealing with that and customer support since then.
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As I said, this was not my first time dealing with an odd checkout glitch as I described. Much earlier this year I had another order do the same thing after canceling and then immediately paying via paypal, and at the time thought it was because of using an outdated browser. Bad on my part I know, but since then I have not attempted to checkout or do anything other than browse GoG passingly unless using an up-to-date browser since then. It is because of this I believe the glitch was likely caused by my canceling and then quickly completing the order immediately after. I cannot prove it, no, but that is my expectation.
Of note, this first time I encountered this glitch I handled it slightly differently. When I found the two orders in order history I clicked 'Complete this Order' on one of the unfinished orders. To my surprise I found it took me back to the cart to pay. I did -not- follow through as I had already received an e-mail receipt from paypal. I went back to the order history and found that the price of that order had increased (due to a sale having ended between my checking out, and clicking 'Complete this Order'.
Not sure what else to do, I took screenshots of both orders, and decided to watch and wait and see what would happen. The other order would soon vanish, and the order I had clicked on would complete (though the stated total continued to show the post-sale increase), and the other order would vanish. The only noteworthy thing about the completed order is that it showed no ability to request a refund.
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That's all the relevant information I think I have for now. I have not mentioned the old incident to support, as I did not want to inundate them with far too much information. Hopefully this can give some insight for people wondering what might have caused this, and help clarify some things I may have been unclear about initially.
Thanks again for all the replies.