You can do it for less than $900, easily.
The Mini P180 is a mATX case, but it's the size of a standard full size ATX tower. If you want something smaller you might think of some other suggestions... Solo, TJ08 or 3480. I have the 3480 and you'll have a tough time making it "silent" though you can get it pretty quiet.
I would suggest i3-530 and whatever decent motherboard Newegg is bundling it with for an extra $10 off. They'll all perform with 10% of each other, it's more about what you need out of the board that matters. Skip the i5-6xx series, they perform only marginally better than the i3-5xx. If you want to move up in processing power the next best step is the i5-7xx series, but these will be hotter and harder to cool quietly.
CPU Cooler - try the Scythe Mugen. It's top 5 rated and the cheapest of the bunch by a decent margin. You can run it passive (no fan) if it's near one of the case fans on an i3. For the i5's you can use the included fan to keep the temps down.
GPU: look for the Powercolor 5750 passively cooled. There are two of them, either will be just as good, one of them (Go Green!) stays under 75 watts peak power if you are concerned about it. It will actually do you pretty good, even with modern games.
Any decent quality 380 watt or higher power supply will be more than enough. Make sure it's 80+ Bronze certified. Both of the Antec cases I mentioned above come with one that will do you just fine. If you choose to go i5-7xx then you may think about 450 to 500 watts for your PS.
RAM - whatever the cheapest 2 GB or 4 GB DDR3 Dual Channel you can find is. If it's Win 7 32 bit then stick with 2 GB. If it's 64 bit then get 4 GB. Any more is pretty much a waste for casual gaming.
I'm guessing all of the above will run you in the neighborhood of $600- $650. Take the remaining $250 - $300 and get in on the SSD thing right now!
Very good! I differ with you only on one point. It's a pain in the rear to add memory to existing memory, especially when you are trying to match what you bought and installed 2 years age. In addition Win7 has reports of chewing up and overrunning its memory. I don't know if extra memory will help, but it won't hurt.
Get least expensive memory, but get some extra. In this day and age 4G is almost too little. Get at least 6G.