I've been building quiet/near silent computers for a very very long time. Generally speaking going completely fanless is impractical and unnecessary. Even if your CPU stays under its rated limit, the rest of the components such as South Bridge and RAM will get quite hot because they expect to have some airflow to help keep the temperature down. I've had DDR2 ram error out on me because it got too hot. Heat kills electronics, especially in the summer and doubly so if you have hot summers and no air conditioning.
A much more practical approach is to use quality components with several very low RPM fans. I have very acute hearing and my personal threshold is about 700RPMs. Any fan at or below 700RPMs is almost inaudible when tucked inside the case under the desk - I'd only be able to hear that in a really quiet room and only if I specifically listened for it. 500RPMs is essentially inaudible for me. I built my GF i7-4770K computer that has four fans all running at or under 500RPMs and her computer is essentially silent, her monitor is way louder than the actual PC because of the backlight buzz.
My recommendations would be to:
1. Buy quality built silence oriented case
2. Fanless power supply, this is important because all fans become noisy after a while and power supply fans are often of lower quality and they're also impossible to change without voiding the warranty
3. Quality CPU heatsink, my favorite has always been Scythe Ninja, it has been the best performing heatink for low airflow application since it originally came out, the latest version is 4 and according to
silentpcreview review it provides excellent cooling with minimum airflow and is still very inexpensive as far as heatsinks go.
4. Quality fans, my favorites lately have been 500-800rpms Scythe Gentle Typhoons (AP-11 is 500RPM and AP-12 is 800RPM), and Scythe S-Flex fans with Sony S-FDB bearings (SFF21D runs at 800RPM). To date I have not found a fan that is better than these, the only problem is that both of these fan types are discountinued, so you might have to buy something else in the 500-800RPM range. If you get the 800RPM models you can undervolt them to run slower. Noctua fans are generally also good if you can find version in the 500-800rpm range.
5. Get SSD if you can avoid getting mechanical hard drive.
6. Use onboard graphics if you can, since you said you'll be using the PC for multimedia/coding I doubt you'll need PCIe videocard unless you're coding for CUDA or something.
If you follow the above guidelines, you can build computer that is essentially silent and you won't have to sacrifice any performance.
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