Q1. The temperature remains constant because the amount of heat energy generated by the lamp (incoming heat flow) equals the amount of energy radiated into the surroundings. If the incoming heat flow is higher, the lamp will heat up, if it is smaller than the radiation into the environment, it will cool down.
Q2. The same process is involved here. The sun heats the water up. The water heats up the surrounding air. When these flows reach equilibrium, the bottle stops heating.
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