eAccelerator is a free open-source PHP accelerator, optimizer, and dynamic content cache. It increases the performance of PHP scripts by caching them in their compiled state, so that the overhead of compiling is almost completely eliminated. It also optimizes scripts to speed up their execution. eAccelerator typically reduces server load and increases the speed of your PHP code by 1-10 times.
I don't see how this will affect Firefox and I don't understand why your host would think so. I would move if they can't answer a simple thing like this with any degree of acuracy.
For your first question, it's shouldn't matter. Your site looked fine to me FF 2.x as well btw. So long as they meet the minimum php and mysql requirements, you won't encounter any difficulties.
For your second question regarding WPMU, make sure they offer Vhost if you are going to use the sub-domain option which is highly prefered and seriously recommended to avoid hassle. Certain hosts like godaddy.com will not give you vhost even if you ask, so you need to ask questions before you sign up for a new hosting accont.
Vhost is generaly not enabled by default, but a good webhost will enable it on your domain when you ask. This is a simple Apache configuration that can be enabled within minutes of recieving your request.
A typical vhost directive looks like this:
Hope this helps and good luck on your search for a more reliable host. I use Surpasshosting (www.surpasshosting.com) myself and they have been very accomodating for me. Reall good deals on the packages, you should at least check them out.