>>2
Truth. (for the most part)
>>1
Complete silence for modern components comes at a price. Solid state drives will be silent but hardly obtainable "on the cheap". And a roomy standard HDD will still make some level of noise. Only passively cooled components can be completely silent. If there are moving fan blades as in active cooling, there is noise; no matter how silent their manufacturer claims them to be.
I suggest you give up the notion of a building COMPLETELY silent computer on the cheap. You may achieve a relatively quiet system at a moderate price if you scale back core specs in favor of passive cooling. And don't forget a passively cooled PSU.
Lastly, "slow" is a relative term. Linux will run fine on even the most ancient machines. Get yourself a tiny distro, convert older hardware to passive cooling and you're golden.