Tom, you've asked a lot of questions, so I'll try to answer them one at a time.
1. Wireless headset - What model phones do you have? Plantronics also makes nice headsets, and has a section that recommends headsets based on desk phone model. Also, there's the Mitel integrated coredless headset/handset available for the 5330, 5340, & 5360.
2. Are you using the internal voicemail or an external system? If your using the internal, it is possible to have users setup their mailboxes and still have the default greeting. Do a search on the controller's help search for "Mailbox Greetings", and by using the Administrator's Mailbox of 9999, you can change the greeting setups of mailboxes. Are you sure that those MWI's that are flashing aren't callback messages from other extensions as well? If you're using the internal voicemail, I'm not sure what network settings you made that caused the phone system to not be able to talk to voicemail, but in order to send voicemail messages as emails, the system needs to be able to use your email server for smtp relay. Are you sure that the phone system can see your email server?
3. What software version are you using? There are two different features that you could use. The first, direct page, puts the caller over the other person's speaker, and the recipient has to pickup the handset or press the Mic key to answer. The second, Direct Voice Call, puts the caller over the recipient's speaker and allows them to have a speaker phone conversation. This second one is a newer feature though, so your system may not offer it.
4. Again, if your using the internal voicemail, there is no direct way to do this. You have to create either phantom extensions, using the format of *ext, and then call re-route always these phantoms to the voicemail pilot number. Then you can transfer callers to those *ext to send the person directly to the extension's voicemail box.
5. I can't offer a suggestion on this one.