Hmmm... I'll try to help, if I can. For problem 1), we need more information about the email account. What kind of access you have, exactly what the phone does and doesn't do (ie - good description of the error). Even then, you might have to contact your work IT dept. to get more help.
For 2), that seems like a T-mobile specific issue. I'm AT&T so I can't really comment. :(
For 3), you might check out
SPB Time. I think that might do some of the things you are after.
For 4), I *think*
PocketInformant might do that. It also does MUCH more than that as well (hence the significant price tag). However, it is great software that I highly recommend. I own it myself.