I have an old Antec Sonata which I'd like to change to a new one. Fractal Design Define R3 seems like a good and cheapish choice.

1. Is it suitable for semi-silent computer? There would be semi-fanless PSU (Super Flower Golden King Platinum+ Pro 500W), most likely SSD drives.
2. Are the default case fans quiet enough at 5V?
3. Does the case vibrate badly?
4. Is the case suitable for a passive PSU?
5. How easy to decouple HDD resonance from the case?

Any other case recommendations?

It think it's still the best option in its class, though the key to a quiet system certainly lies mainly in its components rather than the housing. Anyway the bitumen based dampening material on the panels is excellent, probably the best there is as far a s prefitted dampening goes.

Any case with the PSU in the bottom position is a good match for a passive/semi-passive PSU, but to improve things further I think it's best to avoid an overpressure setup. Because with overpressure heated air will find the PSU area easier. I noticed this effect with my semi-passive Seasonic where having front fans helped overall tempeatures a little bit but the PSU fan was reving up more oftenly. I now run my Define R3 case with just two exhaust fans, having blocked most of the little auxiliary holes in the case to have most of the cold air enter through the main front intakes and the PSU area. Fanless PSU's have holes on their top which improves that setup further.

The default fans in the R3 are 1300 RPM units and run fairly fast even at 5V, I'm guessing around 800-900 RPM. I've replaced mine with Scythe Slip Streams but most users would probably find the default fans okay at 5V for a "relatively quiet" system.

The drive bay grommets aren't very good for drives than vibrate more than typical 5400 RPM drives. Don't tighten the screws too much since that counters the effect of the buffers, or better yet don't use screws at all to fasten the drives to the bays in case the drives vibrate a lot. I have three Samsung HD204UI Green drives in my case and use two not too tightly fitted screws per drive for fastening. I did a test adding a relatively strongly vibrating Samsung HD103SJ drive sitting upside down(in the bay) and secured with straps rather than screws, and then the vibrations were not a problem. Its motor whine was still too much for me so I removed it, but the test showed that 7200 RPM drives could be used as far as vibrations go.

What system do you have?
What needs to be cooled (actively)?
How loud are other components like graphic card cooler and cpu cooler?

It will be a Ivy Bridge setup with most likely 2 x sdd + 2 x storage hdd. PSU will be very silent. Even now my Antec Sonata would be quiet without that damn hdd resonance. I hate it. I'll replace the GPU fan later on. It is also possible that I'll use only the IGP of the CPU. We'll see.

Antec Sonata would be a good case but I'd love to have proper USB headers and a case with PSU located in the bottom.

My big brother's stock fans at 5 V ran very quiet. They were practically inaudible during daytime, though he lives in the center of Copenhagen. For a semi-silent system, they should suffice.