I don't have technical answers, but my phone is provided by my company, and it uses WP7 on an HTC Arrive. I've been using it for about 6 months now. For your questions...
--Email is push and integrates well with Exchange Outlook--contacts, calendar, email, etc.
--Also integrates with Facebook, Windows Live, Xbox/Games for Windows, etc.
--Gmail support seems pretty good, although I haven't tried looking into my archive messages using the phone.
--I checked and don't believe it has hotspot functionality for my phone.
--There are task management apps but no idea about sync support...one Remember the Milk client is here http://julianapena.com/projects/winmilk/
--Don't know about VOIP or SIP.
--With no user-facing camera I'm just gonna guess that video calling ain't happenin' on my phone.
The huge Mango OS update will be later this fall, and you may want to wait to see what that brings before pulling the trigger. So far I'm pretty happy with the device. The OS is pretty impressive, but MSFT definitely put some aesthetic UI aspects a bit above functionality.
WP7 doesn't have close to the number of apps as iOS or Android, but realistically I only use a handful day to day, and I can't go crazy with connected apps as I have to watch my usage amounts.