If the starter is spinning the motor fast (ie just like it does when it’s warm) the starter is fine, the oil isn’t too thick, it’s either getting too much or not enough fuel.
My guess would be crappy ignition components. With the cold and the extra fuel, your spark might not be strong enough to ignite the mix.
Don’t crank the snot out of it either, unless you’ve got the pedal to the floor. The ECU will just keep dumping more and more fuel in.
A nice high output coil won’t set you back much, so why not throw one in?
I had the problem last year and the only thing that would work was that on a failed start I would take out the fuel pump fuse, restart let stall, put fuse back in in and it would start up no problem. A coworker of mine used to tow for a living and found it ery common on Nissans and Mazda to do this.
