Re mixture, what you are doing is correct. I raise the issue because many pilots who are used to just leaving the mixture full rich for everything. Even the sea level pilots should start to lean in the climb above 5000 ft in order to get the best rate of climb.
With respect to carb icing.
1) You can get carb icing almost anytime even in quite warm temps if it is very humid
2) The first symptom of carb icing is a uncommanded reduction in RPM (fixed pitch prop) or MP (constant speed prop). Carb heat should go on immediately if you think you are losing power. The next indicator of carb icing will be the engine will start to run rough due to an excessively rich mixture. Delay getting carb heat in now is serious as the engine could be only seconds from failing completely
3) carb icing will be most prevalent at low power settings because the throttle butterfly will be almost closed causing the maximum pressure differential in the carb. This is why I commented about carb heat before takeoff. For example you get carb ice on the taxi out clear it on the runup but then wait at the hold line at low power before you can takeoff meaning the ice can reform.
4) The runup carb ice check had two components. First that the carb heat is working, you check that by observing an RPM drop after applying carb heat. Second that there is no carb ice. If there is you see the RPM initially drop but then rise again as the ice is melted.
5) if you are inflight and you miss the initial uncommanded power reduction and are now at the rough running engine stage when you apply heat the engine will initially probably run even worse as it eats the melting ice. The temptation is to turn off the carb heat. Don’t do this the engine will clear and pick up power in about 15 seconds.
6) if you have to run with carb heat on you should re lean as carb heat enriches the mixture