WiFi disconnects when iPhone is locked

The answer to this is that the wifi disconnects after a set period in sleep mode only when the device is on battery power, but reconnects immediately when waken. However, if you’ve got Mail set to automatically check for new mail, it will continue to do so.

If you've got it plugged into an outlet, it will remain connected to wifi even in "sleep" mode.

Without hacking the phone, no.

Not very long at all, with the screen on all the time, it's gonna suck your battery dry very quick.

However, here's an idea, don't know if it will work, but might be worth a try, you can check in your router to see if the phone is still connected when in sleep mode:

Attach one of those large spare battery packs I see advertised all over the place. The phone "might" react as if it's plugged in...no guarantees here...

You should look into using Fring with Push Notification - it is a free app in the iPhone store and it integrates with most IM clients. You can turn Push notification on and get notified when the phone is asleep - over the data network. If you are in the US, I assume you have an unlimited data plan - if you don't have an unlimited data plan, Push notification doesn't use that much data.

It may be a better option.