What kind of form is אי? Where does it stem from? You can't use "לא" here?
You can't use לא. I think there is umfortumately no direct translation of אי to English. It's close to "don't", but it is only when applying a general rule, and not a specific instance. Hope this helps :)
I had thought that אפשר was closer to the social acceptability end of the spectrum than to the physical possibility end, so the English glosses provided here ('can' 'possible' etc) don't have their centre of gravity where I would have expected it (they seem nearer to questions of physical possibility than of social acceptability).
Could one say here something like it's 'not allowed | acceptable | permitted' ? Is אפשר somewhere between יכל and מותר ?
ll אי אפשר לעשות את זה means "It's impossible to do it" or "It's not possible...". מותר is allowed/permitted, and you can use מותר לעשות את זה to say that some action is allowed. The negation of מותר is אסור, which is used similarly.
If something is not socially acceptable, you could perhaps use נהוג or מקובל (negation with לא for both).
When is it used?and what is its link to the word אין? And does אי here mean: Nobody?
Just a slightly written 'not'. "efshar" vs. "'i efshar" -> "possible" vs not "possible"
Ok and is this form really used in everyday speech? Or is it more common to say: אף אחד לא אפשר?
I think it goes parallel to our Arabic " غير " ..
As in غَيرُ ممكنٍ = אי אפשר
This subject (modals) is an open invitation to idiomatic expressions. Could this be translated: You can't do that, It can't be done, it's impossible to do that? Plus a few others.