Rather than use a spring compressor or vice, use a threaded rod with a large washer on each end as your press. Tightening one nut (most conveniently the one on top) lets you keep the rod aligned in the head tube while gently and evenly pressing the race on.
I don't have any suggestions for removal. A friend and I replaced a headset at home once, but it's beenalmost 30 years ago and I really don't remember the details. We might have just used a hammer and screwdriver or chisel to remove the races...
Is the old headset so bad that you can't just ride the bike to the co-op one last time?