Yes, but "otwórz" is 2-person-imperative so in order to not make it sound as a command, you should lump a "proszę"(please) into it somewhere(options that sound natural to me are "Proszę otwórz mi okna" and "Otwórz mi proszę okna") – alternatively you could not use imperative but conditional instead, "Otworzyłbyś/Otworzyłabyś mi okna" or for "super-polite" version use modal verb móc(can/be able/be allowed) and proszę: "Proszę, mógłbyś/mogłabyś mi otworzyć okna".
"mi" is just dative of "ja", so you use it (mostly) when you are indirect object of the sentence – "Co mi zrobiłeś?"(what have you done to me?) and so on. As for English, case system generally only work for Pronouns and those don't generally denote Dative(with the exclusion of 'whom'), but:
"While the dative case is no longer very common in modern English usage, it survives in a few set expressions. One good example is the word "methinks", with the meaning "it seems to me". It survives in this fixed form from the days of Old English (having undergone, however, phonetic changes with the rest of the language), in which it was constructed as "[it]" + "me" (the dative case of the personal pronoun) + "thinks" (i.e., "seems", < Old English thyncan, "to seem", a verb closely related to the verb thencan, "to think", but distinct from it in Old English; later it merged with "think" and lost this meaning).
The dative case also survives, albeit rarely, in the ethic dative, used to express one's interest in a matter. This only occurs with pronouns. For instance, in the phrase, "cry me a river," "me" is used not only to express the recipient of the action but the form is used sarcastically in American English to express the speaker's disinterest in the action.
In Modern English, an indirect object is often expressed with a prepositional phrase of "to" or "for". If there is a direct object, the indirect object can be expressed by being placed between the verb and the direct object. For example, "He gave that to me" and "He built a snowman for me" are the same as "He gave me that" and "He built me a snowman". Here, the object pronoun "me" has the same function as a dative pronoun in a language that distinguishes accusative and dative cases."
Yeah, but what i wanted to know actually is: Does polish have so called "ethic/ethical dative"? Perhaps "Prosze, otworz mi okna" is not a good example. For an example: Jak sie czujesz? / Jak mi sie czujesz? Jestes zdrowy? / Jestes mi zdrowy? Does polish use that word to show ones interest in a situation? Couldn't find on google, so i guess the sentences with "mi" are wrong?
There is an interesting article about this:
To summarise, "let's" can be expanded to "let us", but not all "let us" can be contracted to "let's". However, in modern English if you can contract "let us" to "let's" you almost always will. The most common exceptions seem to be religious phrases "let us pray" and "let us rejoice" and phrases with "not", e.g. "Let's not do it" = "Let us not do it".