There are two ways to use a fan in a computer. The first is as a cooler fan, where the fan forces air directly through the cooler. The other is as a case fan, where the fan moves air into or out of the case.
When I say "cooler fan," I'm referring to the first type. Anything else is the second type.
When I say front fan, I'm referring to the fan mounted on the front of the case as an intake, located behind the front door of the R5. There are two slots where you could install a fan. The case comes with a fan mounted in one of those two slots.
When I say rear fan, I'm referring to the fan mounted to the back of the case as an exhaust. It's located behind the CPU cooler, right next to the rear IO panel. This is the one I had to disable to achieve sub-20 dBA levels.
When I say cooler fan, I'm referring to any fans that mount directly to the CPU cooler. There may be up to three, depending on what cooler you have. The NH-D15 comes with two. The NH-D15S comes with one. The be quiet! Darl Rock Pro 3 comes with two.
What mic are you using? You may not need to worry about the computer noise as much as you might expect.
Regarding be quiet! cases, they're very good. They're also very expensive for what you get. The best of the be quiet! cases achieve roughly the same noise levels as the R5 when using air cooling. The be quiet! cases are better if you use a water cooler, though.
Regarding PSU noise, be careful. The PSU can easily dominate overall levels if you aren't paying attention in a quiet build. I'd personally recommend the RMx series or a fanless option from Seasonic.
Regarding water coolers, again, be careful. Most are designed with gamers and overclocking in mind, not silence. The only real exceptions to that are custom loops, the x61/x62, and the Celsius S24. Even those are difficult to design a quiet build around, though. Water coolers require you to have a more open case than air coolers. The openness means that noise can radiate from the computer more effectively, thereby negating most of the noise benefits you might otherwise expect.
Regarding the number of fans to run, you only really need one. Two is better, as you don't have to worry if one of them fails on you. The second fan is totally unnecessary for cooling in your build.
UPDATE: For reference, the build I've been discussing on my end uses an i7-4770K and a GTX 770. It also only uses one case fan, and has been running well for several years now.