Help me pick

A 430W PSU would be fine, but 500 will cover you for most single-card setups in the future. Corsair GS500, $50 after MIR and $60 before: http://www.newegg.com/Product/Product.aspx?Item=N82E168...
If the couple of dollars matter and expandability doesn't, the Corsair 430W is currently $17 (!) after rebate and $37 before: http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Choose the case by stylistic preference. Just about all cases over $50 will perform perfectly well (check here to make sure), and many under that price will too, though that's not as assured. Just trawl the Newegg ATX section and see what catches your eye. Some popular models are the Rosewill Challenger, the Cooler Master Storm Scout, the Antec 300 and the NZXT Phantom.

From what I've read on Newegg for various Corsair PSUs, half of the people aren't satisfied. A lot of people say that their PSU died early on, or that they start to fail days after purchase. Do you know if there is a PSU that certainly won't do that, and that is compatible with the AMD Phenom II x4 BE that I've picked?

For a system running with a single GeForce GTX 560 Ti graphics card NVIDIA specifies a minimum of a 500 Watt or greater power supply with a combined +12 Volt continuous current rating of 30 Amps or greater and with at least two 6-pin PCI Express supplementary power connectors.

It's not the power supply's wattage that determines whether or not it's able to properly power the system with a specific graphics card. The most important power supply specification is its combined +12 Volt continuous current rating.

I tend to stay away from recommending Corsair's budget Builder Series CX power supplies. The Enthusiast Series TX, Professional Series HX and Professional Series Gold AX Series are better depending on which model you're looking for.

What is your power supply budget?