I just checked the dimensions of the case. I don't know why newegg.com called it a mini tower case. Based on the actual dimensions and the fact that it can accomodate a standard atx motherboard it is closer to an atx mid-tower case. What Lian Li did was shorten the height a little bit. That's why the power supply is installed in the bottom front of the case.
Anyway, a standard atx power supply will fit inside that case.
Corsair and Seasonic are two of the brands that have a reputation for high quality power supplies that consistently earn high marks in technical reviews. They are reliable, stable, and come with a 5 year warranty. Some of the newer models come with a 7 year warranty. Lately we've been seeing a few other brands offering some high quality units. One example would be the Antec Earthwatts series which is a major improvement over Antec’s older psu’s like the Basiq models.