Translation:Can I open my umbrella?
Used interchangably but there is an actual difference. “Can” implies the ability to do something. “May” implies seeking permission. Think of it this way. “Can i put up my umbrella?” would parse as “do i have the ability to put up my umbrella?” Because the response is often “yes” it implies permission but in reality the response would actually parse as “yes you can because you have functioning arms and hands” “may i put up my umbrella?” parses as “do i i have your permission to put my umbrella up?” to which the answer “yes” parses as “yes, you have my permission to do so” This distinction is mostly observable when you ask someone to do something for you “can you hold this for me?” which parses as “do you have the ability to hold this for me?” You certainly wouldn't ask someone “may you hold this for me?”
Yes, there is.
~てもいいですか is strongly asking for permission. You'd translate this as "May I...?" or "Am I allowed to...?"
~ていいですか is more like "I want to do..., is that OK?", the speaker is expecting an affirmative response here. I think it could even be translated as "Should I...?" in English (but I'm not a native English speaker so I might be wrong).
Essentially what the も does is putting a very strong emphasis on the permission part: 傘を差してもいいですか literally means "Would it be OK even if I opened my umbrella?", with も representing the "even" part.
Not from the south here, but while I wouldn't normally say "I'm putting the umbrella up", I'd say "I'm putting the umbrella away" but I'd understand that "up" meant "away". They're interchanable in meaning, even if I wouldn't normally use "up". I've certainly never heard of "put up" as any kind of synonym for "open".
that's accurate for sure, and those should be acceptable translations. but without any additional context, it's most likely that the speaker is referring to their own umbrella, and so the most natural translation is probably "my". I'm assuming Duolingo is trying to get us used to inferring context
Japanese used いいですか quite often. いいですか is in "formal speech" format often used when talking to people the speaker is less close to (as opposed to "casual speech" which is used among family and close friends). Casual speech in this case would かささしていい? with a rising intonation. が, を, に, and へ are particles more often than not omitted in casual speech.