Yes it would in a lot of dialects of Swedish. Not everywhere though. But over all, yes.
rs becomes ʂ
This occurs for other combinations as well:
rt becomes ʈ Like in ärta (pea) it's rather äʈa
rn becomes ɳ Like in barn (child/children) it's rather baɳ
rd becomes ɖ Like in lärd (lettered/scholarly) it's rather läɖ
If you want to listen to the sounds you can find them here (under Retroflex): http://www.yorku.ca/earmstro/ipa/consonants.html
Again, it's not like this in every dialect of Swedish but it's like this in standard Swedish.
Yes, you are wrong. All verbs in Swedish belong to different verb groups, the most common of which is group 1. If äta were a group 1 verb, it would be ätade in the preterite tense, just like arbeta (to work) is 'arbetade' in the preterite tense. Instead it's a group 4 verb, and becomes 'åt' in the preterite, e.g. Jag åt din citron igår / I ate your lemon yesterday. All group 4 verbs change vowel when they're being written in the preterite tense. You can see some other examples of group 4 verbs here: http://blog.lardigsvenska.com/2010/10/verb-grupp-4.html .
Group 4 verbs are tricky.
Duolingo has many phrases, sentences and questions that you're very unlikely to ask - I think the "point" is that you focus on the forming of the phrases rather than getting stuck on what is actually being said. Also, it means it's hard to just guess, and you actually have to know the meaning of the words and how the word order affects them. This is not anywhere NEAR as bizarre as Duo can get. The Hebrew course has "Who knows where his finger has been?" and later on, Swedish has "What a big sausage you have!"
Sure (people whose Swedish is better than mine may correct me if I make I mistake, but I think I can do this one): They are all different forms of the word "your" depending on what object (a table, a plate, a lemon) etc. is being referred to. The important thing to remember is that between these three, it doesn't matter WHO owns the object, just what the object is. I believe that you might use a fourth form of your, being "era" if it were multiple people doing the owning. But nevermind that for now, we're discussing din, ditt and dina. Now I've put them in that order deliberately because to me it makes the most sense. "Din" is like to be the most common, because it refers to COMMON GENDER nouns. That term is pretty unhelpful, because it has nothing to do with sexual gender or anything else, it's just a CLASS of noun. I have found it helpful to think of this class as "en" words. Words that have an "en" in front of them if you mean "a car", for example, "En bil." THE CAR would also be "Bilen". Some of thousands more examples include en citron, en jordgubbe, en sköldpadda which become citronen, jordgubben och sköldpaddan when they take "the" in front of them. So you see the pattern, "en" words end in "en" when they take the definite form. They also become DIN citron, din jordgubbe, din sköldpadda etc.
"Ett" words are the next class. They are less common (I believe about 3 in 10 nouns are "neutrom gender", but again, I just think of them as "ett" words). For example, "ett hus, ett djur, ett barn" become "huset, djuret och barnet" (the house, the animal, the child). So they are "ett" words and have "ditt" in front of them if you're referring to "your house, your animal, your child".
"Dina" is the plural form. So it's used if you're referring to someone's plural objects, e.g. "your shoes" = "dina skor", as opposed to "din sko" (your SHOE, singular). I should say, same goes for MY shoes, my child etc. except "min, mitt och mina".
But, with grammar, don't let it stop you saying something. If you're speaking Swedish and you can't remember what form a noun is, just use "en/min/din,-en" form of the word and if you're wrong, either a Swede will gently correct you or you'll pick it up with time. I was speaking to Swede on Skype and said "min hus" and he just said "Mitt hus. Hus är "ett hus, huset, mitt hus, ditt hus" ." and now I will never forget, hus is an "ett" word.
It gets a little bit more tricky when you come to words that don't have a separate plural form (e.g. "barn" means both "child" and "children") though it's still pretty simple because "mitt barn" is "my child" and "mina barn" is "my children" so you get the idea, but that's pretty much it. I hope that helps!