(OP here...) I'll reiterate what I wrote earlier about what works best for me and my Huskee 22T:
1) Start with a clean and dry plug.
2) Push the primer 10+ times, push and release it slowly and be sure to push it all the way in (bare finger works best).
3) Slowly pull the rope until you feel that it's starting the compression stroke.
4) Place foot on splitter and pull HARD. If after 3 pulls it fails to catch, go back to step 1.
5) If the engine catches, immediately position your finger over the primer bulb and prepare to depress it at the first sign that it is starting to die. You mentioned that it will occasionally run for 2-3 seconds, hitting the primer in those cases may have saved it.
Yes, this is one unimpressive engine as far as cold weather starting goes.