Very Cracked Screen Removal

First off, don't be concerned about the amount of heat you use on the display. As long as you're not using a heat gun, regular heat sources can't get hot enough to damage anything other than the screen and the battery. You don't care about the screen since it's broken already, so all you have to do is avoid heating the battery excessively. As long as you're focusing on the edges of the phone rather than the center where the battery is you'll be fine.

Since you've got a lot of broken glass, I'd suggest the next thing to do is to cover the broken parts with clear packing tape. That will help maintain the integrity of the screen while you're taking it off and keep from scattering glass everywhere.

If you bought the Fix Kit then I'd suggest using the Jimmy tool to separate the cracked glass. That, in conjunction with the heat should get you going again. If you don't have the Fix Kit with the Jimmy tool, then I'd say the next best thing would be a curved X-Acto knife; you should be able to pick one of those up fairly easily. Don't worry about using too much force; the frame of the phone is sturdy enough that you won't damage it prying the glass off, and since the glass is already broken you don't need to worry if you break it more. So just be patient and chip away at it a bit at a time and you'll eventually get it off.

One trick I've seen is when you've got the pick or Jimmy positioned on the edge of the screen, drop a couple drops of 90% or higher concentration isopropyl (rubbing) alcohol and let it run down along the edge of the screen. IPA is fairly safe to use on the phone as long as you don't get crazy with it. I've seen it cause distortions on LCDs when too much is used, but it usually resolves itself when the alcohol dries, and besides that you don't care if you damage the display at this point.

Those are all the things I can think of that might help you at the moment, but if I think of something else I'll add it to my answer. If you have any questions about any of this, feel free to ask. Good luck, and let us know how it all turns out.