I'll just add my 2 cents worth since I'm not sure what it is that you're describing. First off, Dell does some weird stuff to save money and scare people off from doing anything inside. F'r instance, I've noticed lots of Dells use a passive heatsink over the CPU, then a plastic shroud over that heatsink that goes straight to the case fan on the back. Saves them some money, but it's non-standard & probly doesn't cool the CPU all that well.
Anyway, I don't have lots of experience with heatsink assemblies, but I was under the impression that the hardware on the backside of the motherboard (if there is any) doesn't bolt to the case. Mine doesn't. My ATX board bolts to the case in 9 places, but none of them have anything to do with the 4 CPU mounting holes. This is on a Pentium 4. The original Pentium 4 heatsink/fan assembly used those four twist-to-bolt down pins, which I hate. If the holes were already being used to bolt down the motherboard, I wouldn't have been able to attach the original heatsink! I soon swapped that heatsink for a Zalman 9500, which uses a plastic frame on the back of the motherboard to support the fan. If the motherboard was bolted to the case right under the CPU, I would have had to remove the brass stand-offs to attach the Zalman. Obviously, I didn't have to do that either.
If you search back thru the posts, you'll find hundreds that talk about thermal paste. I've only been hanging around this forum for a few weeks and have seen several. Reader's Digest version - you should clean off the old and apply new.
Are you moving the Dell power supply over to the new case? Dell pulled off a dirty trick on unsuspecting consumers - they use (or used to, I'm still not clear on this) a proprietary wiring scheme between the motherboard and power supply. If you swapped a mb and kept the Dell PSU, or vice versa, you'd ruin one or both of the parts. If you're not changing the power supply or mb then you don't have to worry about this.
So, my answer to your original question is I'm not sure but I don't think the heatsink/CPU mounts are supposed to attach to the case. You may have some funny issues with the way Dell did things that don't apply in your new case.