I'm making the assumption you are connected to WiFi when using the client (and not VPN'd in over 3G). The iPhone behavior when connected to WiFi is that when you put the phone to sleep, the phone turns off your WiFi to save battery. Then when you wake the phone up it will reconnect to your WiFi. When connected to WiFi I do not believe you will receive PUSH notifications for email. I think the only way when connected to WiFi to refresh your email accounts that are using PUSH is to either open the Mail app, or select the refresh button if the Mail app is already open.
So when you turn on shake to lock with the Mobile 8 client it enables you to lock the screen without putting the iPhone to sleep, which in turn keeps your WiFi connection active and disables PUSH notifications for Mail.
I have noticed the behavior you mentioned where the client unregisters from CUCM if you close it. Right now I don't beleive "multitasking" with 3rd party applications is supported on iPhone OS 3. However I believe this functionality will be in OS 4 when it is released and if so you should be able to have the Mobile 8 client running in the background so that you can check Mail, or use other apps while it is running.