"Why don't we open the window?"
@BJCUAl has already mentioned above まどをあけましょうか？ should be accepted; So should be まどをあけましょう which is affirmative but also a suggestion. There is no direct negative form for the indefinite form ましょう, but indefinite form of the negatve form あけませんでしょう; It is not used in a suggestion as far as I know but it is not related to appropriateness. I have finished the Duo course and I do not remember seeing appropriateness covered.
I'm using another source to learn Japanese, which uses the word 「開いています」 to express "it's open" or "are open", as in「お店は開いています」 (omise wa aiteimasu / the stores are open). Now, I know what the usage of the 'て form' together with「います」implies, but, since it's the same Kanji, the difference between「開け」and「開いています」seems to be different from that of, for example,「食べる」and「食べています」. Could someone please explain me why is it?
They have a similar meaning but are conjugations of two different verbs,
開いています is the continuous/stative form of 開く which is intransitive, "to open", as in "The door opens" (future/habitual) "The window is open" (stative/continuous)
Its polite ます form is 開きます
開けません is the polite (negative) form of the transitive verb 開ける "to open" as in "I open the door"
Its continuous form is 開けています "opening"
Two interpretation of this question, especially when context is not specified. 1. Real question with "why".....because the AC is on. 2. "let's open the window"....instead of the door.
The given translation somehow does not seem to be right with either of the above interpretation.