One thing to note is that your primary goal is size. This can mean multiple different things. You mention the overall volume, but in a practical sense, that's a pretty meaningless number.
Something like a Coolermaster Elite 120/130 is fairly small in terms of volume, especially considering what you can fit inside it (you won't go much smaller and still be able to fit a full sized ODD and PSU) but unless you have a low/narrow kind of gap you need to fill in a desk, it's not all that small in a real sense.
One of the lowest volume cases you can buy is the EVGA Hadron Air, which manages to be very small due to the non-ATX PSU. If sheer cubic capacity is the goal, then it's likely to be hard to beat, though there are compromises.
Small for most people, in the practical sense, comes entirely down to the footprint of the case, as a lot of small cases are going to be placed on a desk, where you aren't height limited. In this instance, the Silverstone FT03-Mini and Silverstone RVZ01 are both pretty good, in different proportions, though both have their issues too (slim ODD, SFX PSU, limited cooler options etc).
Performance these days isn't really dictated by size but by your wallet. High end Z87 ITX boards aren't far off their ATX brethren, and the in game differences are likely to be near to zero. In a lot of ITX cases, graphics cards sit alongside the edge of the case, so have their own air supply, meaning that they can often get better temperatures than in a massive ATX tower with a silly number of fans.
If you are using a standard ITX case skewed towards gaming builds, then you will almost undoubtedly have 250-300mm of double slot graphics card space, which basically means you can fit any graphics card on the market, though obviously each case is a bit different. This means there isn't really any advantage in hunting for shorter cooler graphics cards (like the Asus GTX760/GTX670 Mini or MSI GTX760 "ITX"), and once you go above this level (GTX770/GTX680 etc) you are limited by the length of the PCB itself, so even a custom case wouldn't be able to cut down much in this area.
Noise is generally incompatible with performance, as basically any choice you make towards silence will have some sort of impact on the airflow of the case. This includes things like using solid panel variants (rather than mesh/vented panels) or picking components which are, for example, fanless. There are several you can do though -
Make sure to have some level of control over fans, either fan controller or through the motherboard.
Buy quality fans. Good fans are usually quieter due to higher quality bearings, plus can be run at lower speeds.
Be prepared to swap characteristics as required. Most systems can easily be made "silent" by running fans at lower speeds when maximum performance isn't needed, and a lot of graphics cards come with fairly aggressive fan profiles to make them look good in performance benchmarks, something which can be tweaked really easily.
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