If you are plugged into a 15 amp outlet and the connector gets hot, you are trying to pull more than 15 amps though it, which you should be careful not to do when, when plugged into a 15 amp circuit. The breaker should trip when you exceed 15 amps, so that breaker should be checked. As far as leaving it plugged in while stored, I always leave mine plugged in, and use/store in use. I saw a comment that the converter is not the best, but I have had no issues with the one in mine. It keeps the house batteries charged, and cuts back when they are fully charged, exactly what it is supposed to do.
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