I would recommend letting it run at least until the revs come down to around 1000. This will at least get the oil somewhat less viscous.
Logic tells me it couldn't be a battery since it starts initially, although if you have the stock battery, you may want to look at getting a new one since these cold temperatures take their toll on 3 year old batteries pretty quickly.
What octane of fuel are you using and from which fuel stations? I always use 93 from Shell, BP, or Mobil.
An immediate stall when you let the clutch out makes me think that your car is a) lacking torque (for reasons outside my scope of knowledge, but could have something to do with bad fuel and very high air density), or b) being told to shut down by the ECU because something is wrong. Perhaps a knock sensor or an O2 sensor is bad? Does the check engine light come on when you start it back up? I'm assuming no but thought I would ask anyway.
For reference, my car has 21,000 miles on it and the original battery. Only mod I have is a CPE Intake
I'm no expert, and not a seasoned car mechanic, but I've worked on and diagnosed my fair share of cars here and there. Just throwing some thoughts out there.