Well... I was a two stroke mechanic for a couple big name motocross teams, and I am a Factory certified Honda, Yamaha, Kawasaki, and BMW motorcycle, ATV, Watercraft and a Honda Power Equipment Tech for about 8 years.
I also have two world records with
R/C outboard boats (with two strokes). (make sure you watch the whole thing with your speakers on, I built the engine for him.) I am not an expert at much, but its safe to say small engines is my specialty.
Here is my analysis based on what you have given us:
Assuming you have a proper fuel mixture with fresh gas and quality oil.
Assuming you have a decent plug in it with good spark
Assuming you have a clean air filter.
Its cold outside and the air is thin. Its lean.
Thats why its hard to start, and when it does, it revs up (lean fuel mixture), and bogs or dies when at high rpm (lean fuel mixture)
These little engines are about as simple as they get mechanically, but they can also be finicky once you get some wear on the engine which looses its seal. Think of a two stroke as a pump. When the pump (ring gap, crank bearings/seals) start to wear, the pump become less efficient. Thus making the fuel mixture more finicky.
I have a ryobi weed whacker that has got to be 14 year old, and its still stock, but this year she started getting finicky.
Try this:
Your carb will have two spring loaded screws. One screw is the "low speed" needled, and one is the "high speed" needle. They are just what they sound like. The low speed needle controls how much fuel is allowed to mix with the air at lower speeds, and the same for the high speed needle at higher RPMs. If you have a Walbro carb on it, it usually will have a small "L" or "H" designating which needle is which.
The low speed needle is not as finicky when it comes to adjustment, in other words, two full turns on the needle would be a lot of adjustment. The high speed is the opposite, 1/2 turn on the needle would be a lot of adjustment. Got it?
Get it fired up, and start to turn
out (to richen) the low speed screw until the idle gets down to a normal two stroke pop. Once you get that sorted, let it warm up, then turn
out ( to richen) the high speed needle. You should be able to peg the throttle wide open, and hear the engine clean out but not over rev. Once you get to that spot, make sure it will still idle good, if so you are done. If not, just tweak them both a little at a time until they both work like they should.
If you cant get to the needle screws with it all put back together, you should try turning out the low speed 1 full turn, and the high speed needle about a half a turn out, put it together and try it.
Its actually a VERY simple process, but sound like a lot. If you can get to the screws, you should be able to tune it about 30 seconds after its warmed up.
If this does not help, there is a rubberish diaphragm in the carb which acts as a fuel pump, and they sometimes get holes in them from deterioration. You may need a new diaphragm. But I am 95% sure its lean, due to either fuel mixture of crankcase leaks (wear).