If Microsoft\'s method for using IE as a local host for HTA\'s then, can I use any other browser instead?
You can't.
There is a similar Mozilla project named XULRunner, but HTA only works in Internet Explorer - the two technologies aren't compatible.
To make this clear: HTA will probably not work in anything that's not IE. Yes, HTA is a browser control in a window, but it also has normal aplication privileges (i.e. filesystem access, registry, arbirary code execution etc.). When running in a browser, these privileges are denied (for good reasons - you don't want just any webpage to read your files).
So, launching the HTA in a browser will work, but any interaction with the local system will fail, which negates the whole idea. Launching a HTA in XULRunner will also fail, because XULRunner uses a completely different API.
The only scenario that could possibly work is this: a webapp which doesn't use any of the HTA-specific or XULRunner-specific API (i.e. behaves like a normal browser app). In such case, your app might work with HTA, XULRunner, as well as in any browser. Incidentally, this would take away any advantage of using HTA and introduce masive security holes because of the higher privileges; you'd be better off with using Google Chrome or Mozilla Prism for that.
(We've been trying to find a cross-browser solution for some time, and HTA is definitely not it)