No, that is not normal. A Stihl MS 250 smoking excessively and running hot typically indicates an airflow or cooling problem.
The first and most likely culprit is a clogged air filter. A dirty filter chokes the engine, causing a rich fuel mixture, more smoke, and poor performance, which makes it work harder and overheat. Pull the filter cover off and inspect the foam filter. Clean it in warm soapy water, let it dry completely, and re-oil it if it's the oiled type. These filters should be replaced after every 25 hours of operation.
Next, check the cylinder fins and the air intake on the fan housing. If they're packed with sawdust and debris, the engine can't cool itself. Clean them off with a brush or cloth. Never use a high-pressure washer, as the jet can force water into seals and bearings.
If that didn't work: You might have the winter operation shutter in the wrong position. If it's above 20°C (68°F), that shutter in front of the spark plug should be closed for summer operation. If it's open, it draws in hot air from around the cylinder and can cause overheating and running problems.
Heads up: After a long period of full-throttle cutting, always let the engine idle for a minute before shutting it off. This lets the cooling air flow dissipate the heat and protects engine components.