healer said:
If you don't mind, please tell me why a potential form is used. Would it be the same meaning for the same purpose if 涼しくいる is used instead?
I thought I alluded to this in my response. The idea is "we
are able to stay cool." The potential form of いる ("to be") suggests having the capability to be in that state. 涼しくいる without the potential would just mean "to be cool" without the notion of
being able to be in that condition (thanks to the air conditioner).
Most textbooks don't explicitly cover this because it doesn't occur as often as with する and なる but it's the same basic grammatical pattern.
One thing that's worth noting is that to express a similar sentiment with -な adjectives (or nouns), you'll typically get phrases like いつまでも綺麗
でいられる or 友達
でいられる, not に as you might expect if you think it would follow the same pattern as (い-adjective)くなる or (な-adjective or noun)になる. In this case, で is interpreted as the continuative of the copula, suggesting being able to
be in that state. Verbs also use the -て form in a similar way (緊急時でも落ち着いていられる, 安心していられる). You can also get the negative, e.g. ここまで来たら、もう黙っていられない!, etc.
As for your final two questions, I'm not 100% certain what you're asking. Even if you haven't been explicitly taught this pattern, the specific grammatical structures being put together are pretty self-explanatory and comparable to other things you've seen, yes?
For your final question, I'd lean towards saying "yes", though I'm not completely confident because I don't know what you're envisioning. It's worth pointing out that because of the nuance of this expression, there are certain words it will more likely occur with (states where remaining in that state is a desirable thing). That is to say, you're going to see 涼しくいられる, 優しくいられる, 安心していられる, etc., but don't expect to see 蒸し暑くいられる, 気難しくいられる, 疲れていられる, etc. except in rare circumstances, if ever.