Translation:I study with my friend on Sundays.
と can also be used to signify quotations, like after a quote. 「それはすばらしいです!」といった。 "That's wonderful!", (I/you/he/she/...) said. Whereas て can also be used as a connective for some words, for example dropping い on i-adjectives, for example やさしゅい (gentle) and あつい (hot) and adding -くて, you can list several adjectives like they're connected, やさしくて、あつくて (gentle and hot/passionate). This example is from the song モノクロのキッス by SID
Yes, kayou has the implication that someone is attending/going to appointments on a regular basis. It is to iku as tsutomeru is to hataraku. There is an implied sense of obligation as in regularly attending for kayou and longheld loyalty for tsutomeru. It's hard to explain in English!
空様、なし。The 'on' is unnecessary : ) you would say either I will play tennis next Sunday OR I'm playing tennis next Sunday. The latter would be more common eg. if a friend said - what are you doing next Sunday? you would reply - I'm playing tennis (as in I'm (going to be) playing tennis. And you wouldn't need to repeat "next Sunday", your friend already specified that that's the day they're asking about so you don't need to repeat it again.
It's speaking in general terms - as in this is something that you generally do on Sundays. If it meant every week then it would be modified by maishuu no (every week). If you meant I'm going to study with friends this Sunday you could say either konshuu no nichiyoubi (Sunday this week) OR tsugi no nichiyoubi (next Sunday ie. the upcoming Sunday). I would stick with tsugi no nichiyoubi as since Sunday is the first day of the week you couldn't say Sunday this week and mean a Sunday still to come - for it to be Sunday this week you'd have to be saying this on Sunday, so why wouldn't you just say today? Anyway - there's some discussion about this very question above.
No, it can't mean that. と follows 友だち indicating that 友だち is who the speaker is studying with. Hence, the friend can't be the subject of the sentence (or performer of the action) while it's marked by と. Also ともだち is 友達. The kanji that you have - 友人is ゆうじん. It means a very close friend.
This is just a guess but I think often in Japanese whether something is plural or not is left up to context??? I would love if someone who knows more would confirm that or correct me but I think there isn't really a way to pluralise it... maybe if you said something like "I always study with my friend on Sunday" that would give a stronger impression that you're talking about multiple Sundays but I don't think there's a plural really??
Wait, what? "My friend studies on sunday" wasn't accepted, but none of "tomodachitachi" "watashitachi" nor "watashi to tomodachi" were used. "To" is there, but I thought that was being used like it is in "~to i masu" to accent the person being talked about (in this case, "tomodachi.") Is it really meaning "my friend and (me/I)" by "tomodachi to" here, with the speaker's involvement being implied after to ("and")? Can someone clarify this for me?
Why does Duolingo randomly hate Kanji? I've come along far enough with Kanji that I do not use the word banks anymore, but it seems to randomly decide when and when not to accept an answer using all appropriate Kanji. I can never figure it out.
日曜日は友達と勉強します。 Should be an acceptable answer... it's literally exactly what was spoken.