Now if I can just get it to stick!
The older I get, the less things want to stick. 😀
I don't go into Java weekly, but it's set to "Notify me to install updates". My understanding is that, on that setting, it will routinely check, then prompt me if necessary. Am I correct about that. If not, I'll reset to allow. I'm not THAT diligent.
I have Java set to check automatically. When there is an update, a popup appears on the screen. Trouble is, it may happen when you're using your browser. Then you can't be sure if Java created the message, or it's a fake popup the site is trying to get you to respond to. When I see those, I shut the browser down. If the message disappears with it, you know it was a fake popup. If it's still there, it's legitimate. Even so, I still close the message and go to the Java preference pane to ensure it is calling for an update, and then run it from there.
I'm currently up to date for MY system; the caveat being that I am still on 10.9.5 because of a project I need if for. Not great, I know, but I'm upgrading as soon as I'm done . . . though I like it; it's nice and stable and I've come to like the UI better.
The main problem with that is Safari. I think Apple is still releasing security updates for Yosemite, but I'm not sure. When they stop (and I think they relegate updates to the current OS and two back), then Safari and the OS are open to any exploits that show up later. Worse, they won't be fixed. Something to be aware of.
When iOS 7 introduced the flatter GUI, you knew it would only be a matter of time before the Mac OS went the same way, which it did in El Capitan. It looked odd at first, but it only took me about three or four days to get used to it. After a couple of weeks, I didn't even pay attention to the change. In fact, the older design now looks outdated to me.
I can't find any options locally in my browser prefs. Only JavaScript options.
Correct. All Java controls are in its preference pane. Where you have the security level set is irrelevant if you have it turned off for your browsers, per this check box:

Really, Java should be off at all times. If you have a particular site you use where it must be on, turn it on only for that session. When you're done using that site, turn it back off. If it's a bank you deal with that demands you use Java in order to do online transactions, they are a bank that is totally, almost criminally devoid of brain matter. Requiring the use of such a known security Swiss Cheese plugin as Java means you should all of your assets to another bank - immediately.
As far as Java apps on your computer (such as the ROES software I have to use to send client images out for prints), they're no more dangerous than any other app run directly from your Mac. Either it has malicious code intentionally written into the app, or it doesn't. No app, or even the OS is completely safe from a disgruntled employee. Though bigger companies like Apple, Adobe, etc. have engineers who look over all new and reworked code for errors. Part of that job is also to look for things that aren't supposed to be there so it never leaves the building.
It still vexes me however that the router is allowing the redirect. Am I right that that SHOULDN'T be possible under my current circumstances? Any thoughts how that might be?
I don't know what your router allows for blocking, but it could be the URL itself. Let's say you blocked the very long URL you had posted earlier. They do that on purpose. To get by such blocking attempts, those long strings are constantly changed so they don't match your lists. That's one possibility, anyway.