There's UC Browser. It probably uses IE's rendering engine since the big names (FF, Opera) are missing.
You can't have IE in Kid's Corner (the guest mode) so you'll want UC Browser as an alternative. There are also alternative phone dialers since you can't have the default phone dialer in Kid's Corner.
On Windows Phone 8? I've been using Windows Phone since the launch of WP7 and I've never heard of a competitor's browser being allowed, nor of a decision to change this. Windows 8 possibly, but the OP seems to be confused between the two because Windows 8 doesn't run on anything ARM made by Nokia.
Ah, well, people seem to believe that my above post is a troll, so let me elaborate.
Windows 8 explicitly blocks the installation of any browser other than Internet Explorer (+ skins) on ARM. This is not due to a technical reason. This is due to the fact that Microsoft imitates Apple's policy that prevents any browser other than Safari (+ skins) on iOS [1]. While anti-trust rulings have forbidden Microsoft from doing this on PCs, Microsoft claims that ARM-based computers are not PCs and that the anti-trust rulings therefore do not apply. I therefore suspect that Microsoft will continue with this behavior unless a court of justice decides that ARM-based computers are also PCs. Hence my above comment.
And if anyone wonders why we need open platforms for mobile devices, that's a pretty good reason.
You can't have IE in Kid's Corner (the guest mode) so you'll want UC Browser as an alternative. There are also alternative phone dialers since you can't have the default phone dialer in Kid's Corner.