I definitely understand - this is a weird one and I l haven't seen a weird one like this in a while!
If the ISP has swapped the router before, you can probably contact them for the login information for the router. Either that, or you may be able to Google the default login for your router (if you haven't done so) and see if that works (I notice ISPs tend to leave the login at the default).
Regarding seeing different times, yes - it's because I'm in Idaho, USA - and that time will show up to be whatever the local time converts to.
The step like issues definitely *could* be linked. The strange thing to me is that we saw those step-like issues (which I can confirm would cause some serious connectivity problems) in one instance. For the other instances of your outages, it seemed like the only thing weird was that the route had changed to show a small number of hops for one trace. THAT is the head-scratcher here.
Regarding the hops switching, I have to say that I think it's interesting, but I don't have a lot of insight into what that would mean for you. Your ISP would have a better idea - so you might want to tell them what you told me (that the outage happened exactly when the route changed drastically, and that pings are still going through but nothing else is). If PingPlotter is showing green, but your computer says "no Internet connection" - that means that pings are working, but the rest of the traffic isn't (and it would make sense why your ISP said they couldn't see an outage in this case). This is why I'm interested in seeing what Wireshark has to say.
To use Wireshark, just open it up (but do this while you notice the outage, preferably during a time when your computer says there's no internet but PingPlotter is working). You'll be looking at a screen that has a search-like bar, and it probably has a list of a few different network adapters beneath it with some squiggly lines next to it. If you see an adapter named "ethernet" or "wifi" and it has some squiggles next to it, double-click it. If there's only one in the list, or only one thing with squiggles, click that one. Then, this will open up something that just captures your traffic. Let that do its thing, maybe try a Google search or something (knowing it won't work), and keep PingPlotter running. After about a minute or so, go ahead and press the red square to stop it. You can then head to File -> Save As..., and save the file. Send that file to us (but I wouldn't recommend sending it on the forums - feel free to email us at email@example.com with the file attached and ask for Hayla).