Will a Russian citizen with US green card traveling through Stockholm need a transit visa?

I recently flew from LAX to Pulkovo with a layover in Stockholm Arlanda and was not able to go to my connecting flight without going through passport control. I was initially told I cannot enter the airport without a Schengen visa, but accidentally ended up in the Schengen zone when an airport employee mistakenly bypassed passport control and took me there by elevator. Luckily, I was able to return airside without a visa.

I'm returning to the US next weekend and was told by my traveling companion (who recently flew back) that she had to go through passport control and re-enter the security checkpoint even though she was connecting to a non-Schengen destination. She's a US citizen.

Am I correct in my understanding that my US green card will allow me to go through passport control without a Schengen visa? I'm a Russian citizen.

There doesn't seem to be a way to get to my connecting flight airside without going through passport control. My connecting gate was not listed on my ticket and I didn't see a connecting gate on my companion's ticket either.

EDIT:

Arriving flight from LED is in Terminal 5. Departing flight to LAX is also in Terminal 5. Both flights are from gates within F58-F69.