I have a Nokia Lumia 920 with IE 8.1. I can't answer Captcha questions without an updated browser. I have considered updating to IE 10. What should I do? The site suggests Safari, Chrome, Mozilla. Can I even use any of those on my phone (esp. Chrome)?
The best way to solve our problem is: Update your browser, or Change with another ones. I use Lumia 532, and my browser is UC Browser. Until I answer your question, I have no problem with common website/browser problem. Like Error in Captcha, Playing video, etc. Sometimes, captcha error caused by unstable internet connectivity.
WP8.1 comes with IE11, which should support approximately all CAPTCHA systems. However, IE11 support is being phased out of the modern web, so it's possible you've run into something that just doesn't bother supporting it anymore. While IE11 had moderately good standards support at its release, it hasn't received much in the way of functional updates since then (the only notable one that comes to mind is support for HSTS).
Windows 10 Mobile (W10M) comes with Edge, Microsoft's new browser to replace IE. Current versions of Edge support approximately the same web standards as current versions of Chrome or Firefox. While installing current versions of W10M on a Lumia 920 is not supported, it is possible (if you don't mind downloading some "hacking" tools for your phone that will edit its registry).
Note that third-party browser apps for phones are very rarely going to support web pages that the phone's built-in browser doesn't. When you install Chrome or Firefox on a desktop, you are getting a new "layout/rendering engine" (Blink for Chrome, Gecko for Firefox) that has different capabilities than the ones that Windows comes with (Trident for IE, EdgeHTML for Edge). However, on a phone, third-party browser apps are basically just wrappers around the same engine that the phone's built-in browser uses (Trident or EdgeHTML, again). Still, you could give it a try.
Site suggests Safari, Chrome, Mozilla. Can I even use any if those on my phone (esp. Chrome)?
No, you cannot. They are default error pages for IE. And they aren't meant for Windows mobile. Most websites don't make seperate error pages for Windows phone users(unless they have an app on the store).