Why FA20 engine is noisy?
 

Is it because the way it is designed? Or its a characteristics of any Boxer Engines? Does WRX STI (ej20/24) also noisy like this one?

Dual injection, horizontal, over head cam, thinner exhaust manifold material, less sound deadening everywhere, etc.

small tractor engine

How about STI's engine?

Pretty sure it's the Twins which are noisy, not the engines. All engines are noisy, they are just sound-insulated better. But sound insulation adds weight and reduces the raw feeling of the car, so the Twins don't have much.
I'm sure if you up the sound insulation on a Twin to a similar level as a regular commuter you'd get similar noise levels. Alternately, compare a running engine with the hood open across different engines to see if the FA20 is really any louder.

Sound tube, minimal noise insulation to reduce weight.

Same thing. You just hear it more in the Twins because of the dual injection and lack of soundproofing.
Most cars are so soundproof now that they just hide the sound that they all make.
You want to hear noisy take a base model 60s Dodge Dart with steel door panels and dash with no factory carpeting for a spin! The Twins are like driving a Rolls in comparison.

Isn't port injection only added at low and high RPM? Wouldn't then the bulk of operation be direct injection only? I remember seeing a graph detailing the breakdown between direct and port injection, but can't find it now...

I recall seeing that one too. Can't find it either.

IIRC, the car starts on port (maybe because the DI pump needs a moment to ramp up fuel pressure? It's mechanical, right?), and then always uses DI as soon as it becomes available. Port is only active as a complement below roughly 4 500rpm. Past that, it's DI only.

Some Ford engines are the opposite, using a majority of port with DI as a performance complement. Which allows them to idle on the quieter port system.

But as others pointed out, the FA20D's a fairly agricultural engine with a rather high degree of complexity (4 cams & 8 injectors on only 4 cylinders), plus the car lacks deadening and is running on a platform that wasn't created for luxury.

I *think* the EU cars are a little quieter due to a more restrictive rear muffler and more hood sound deadening, though I guess it mutes them more than it refines them.

Yep. And you can really easily hear when they kick in and out.

I love mechanical noises and it gives me a feel of owning a sports car. But they are so loud and every single days I believe I can hear new types of noise. Due to lack of automotive mechanical knowledge, it just scare me and push me to believe something is wrong. On the other hand, many peoples told me the engine doesn't feel like pickup truck engine rather than sports car engine.

They have so much horsepower the sound slices through the engine bay

my 2015 Legacy at cold startup, is louder inside the cabin than when I had my BRZ stock under the same conditions.

sometimes you have to quantify what exactly is "noisy", because context matter.

I have excluded anything related to rattling including cabin. I have compared it to my friends 180SX, my 86 has 3 times louder noise than him.