The usual issue with swapping motherboards into a Dell case is whether you'll be able to connect the new board to the front panel. Dell typically uses a custom connector between the motherboard and front panel. Wiring for that connector is not publicly available so you'd have to trace the wiring yourself.
If you buy the new board you may have to consider a new case too. And you'll need to buy a new license for Windows because the Dell Windows Reinstallation disk won't work if it detects a non-Dell motherboard. And you will have to reinstall Windows because the new motherboard will probably need drivers that are different than those used by the old board.
The I580 should support a Core i5 processor according to the manual, though I don't know about the specific one you picked. Do you have the latest version of BIOS which might have support for this processor?
You should also be able to use up to 8 GB of RAM. The 4 beeps indicate a RAM problem so are you sure it's installed correctly? And do you know if the new RAM is good? Dell PCs are also very fussy about RAM and some brands may not work even if it has the correct specs.
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