Can someone answer my questions about ISPs in easy-to-understand language? [View All]

The non-profit I'm volunteering for owns its domain name.

We rent space in a building that has wi-fi provided by the landlords.

The problem is that a volunteer (now long gone) established an email service with this crappy company that doesn't return emails or phone calls. We can only access our email about 15% of the time, the rest of the time it tells us it doesn't recognize our passwords. It also doesn't forward messages to the Director's Outlook. The email company is also supposed to be developing our website, which it hasn't done so far.

We do pay our own phone costs, but we can't sign up for Bell South email because apparently our building isn't able to be wired for it. I don't understand why they have to wire anything since we have wi-fi.

We need a dedicated address because we have a mailing list of over 500 people coordinating to rebuild the Gulf Coast, including governmental agencies. But with a list so extensive, we've learned that our emails are being sent into spam files, especially with the government emails, and we really, really need them to get our emails. So Yahoo, Hotmail, Gmail, etc. accounts aren't an option.

So my questions are:

Can we use our domain address to set up email through another provider when we're contracted with this crappy company already?

How do we find a provider who can work with our existing wi-fi system?