Welcome.
Struts "appear" to be factory STi. Springs look to be aftermarket and either cut, as you stated, or heated and then slammed.
Intercooler part number search turned up a Swedish salvage yard selling this from wrecked Foresters. They even had pics of the Foresters they pulled them from.
Yes, you are missing the air box intake ram.
If I were to venture a guess, I would say the previous owner had other parts on this car and parted it out prior to sale. Then put on whatever he found and put it up for sale.
A thorough exam would be in order.