Yes the hyper 212 is compatable with socket 1155, you can see this on the details of the item on the newegg link you posted. Yes the I5 CPU comes with a heatsink and fan, this is also listed under details on your link for the CPU.
Tom's didn't have a benchmark for the I5 2405s but here is a comparison between the I5 2400 and the Phenom 955. As you can see that I5 beats that AMD CPU in every test run. Tom's Hardware - Benchmark Comparison
The Intel® Core™ i5-2405s is a boxed processor that does come with the stock HSF (heatsink/fan) that should work for the normal course of running this processor. Now if you are going to overclock, looking for a silent environment or a very small thin form factor chassis then you might be looking to add a 3rd party HSF.
The Intel HD 3000 Graphics are good but not up to most of our gaming standards today.