Over budget by a little... The i3-6100 is a pretty big step backwards, but would get you under budget. If you can stretch the budget a bit further, the i5-6500 is a welcome upgrade for its high base clock speed over the i5-6400.
If you can fit the need for an aftermarket CPU cooler in a sub $400 rig, then sure... include one. Without overclocking, the stock Intel cooler will be just fine.