I run the tape outs to a soundard. Its how I get my mix into the computer for 2 track processing. Its about as common as having faders on a Neve nowdays.
You can do that, but depending on what you call a soundcard you will get differing results. A quality soundcard will help you get a more professional sound.
I run the tape in and out to the line in and out on a soundblaster mp3+. I'm using a berhinger mx602a mixer. That lets me route the playback to my phones/control room out for monitoring without routing it back to the recording side. The mixer has a set of switches that either stop tape in, add it to the monitor mix, or add it to the monitors and mains. Just don't add it back to the mains or you will get either a doubled signal, or very bad feedback.