For your quick transfer for voicemail, check out 1-2-24 and 1-8-26. That should get you going in the right direction for that issue.
For your speakerphone issue, I've run into this before with Elite IPK systems and Dterm phones. Had to go into 4-12 and program a "MIC" key, as the actual MIC key on the phone doessn't have a key-map on the IPK Systems. Not sure what handsets youre using on this Level2 though.
To resolve the DIal 0 to reach the operator issue, you'll want to check out 1-1-46 and 1-2-08.
AS for the VM issue, not sure if you're trying to access subscriber messages with a softkey, or an actual MESSAGE key on the phone. Your integration sounds like it may be off. Depending on the type of mail you are using (VMS, FMS, etc) you may want to check out the switch/PBX settings of the mail. I know in the VMS/FMS there's a page with an integration string that I would always mess up.
Also if the manual you found for the Level2 seems cryptic, you can always grab a copy of the NEC Electra Elite IPK Features and Specs, and also PRogramming manuals -- they're almost identical.
I tried to give you as much info as I could without giving you exact instructions. I'll let you figure out the rest

If you do need more assistance, just ask!