"Shall we go swimming at a pool this Sunday?"
Learning kana definitely has its advantages. Not every word is written in kanji. There are a heck of a lot of kanji to learn - even Japanese people often need dictionaries to read the newspaper! If you can read kana then you can read children's books and if you can read kana you can read anything with furigana. If you can read kana then you know what to look for when you're looking up a new word in a jisho. The order of kana is crucial for learning to conjugate. Kana is helpful if you have a kanji that you don't know and context is not giving you any clues - then you at least know how to read that kanji and can look it up in a jisho especially if you don't have any kanji books/resources.
I think he means that after having learned kana, having the words written in kanji (even if with furigana) would probably speed up the learning process (for Japanese language as a whole).
I personally think that at this point of the course everybody is able to understand kana (at least hiragana), so they can already do all that you mentioned. So putting the words in kanji (at least for kanji up to JPLT N4 and most common words) would help everybody get familiar with how they are written in real life without making them less able to look for words in dictionaries
That has nothing to do with what he said. Nobody said that one shouldn't learn Kana, and I'd be very surprised if somebody got this far in the course without knowing every kana. He merely suggested that more kanji should be used. To be honest, considering the fact that anyone who is serious about learning Japanese probably would know all 80 first grade Kanji by now, duo should at least be showing all of those whilst gradually introducing the next lot, with furigana to address all the problems you took far too long to type out.
I checked out a few Japanese children's books from the library recently and was dismayed that I couldn't read them. Some of what I didn't know was vocabulary, but a lot of it was unfamiliar syntax and grammar. In a gosh darned children's book. I've completed Duo's whole Japanese beta and I still couldn't even read children's books. And because they were written entirely in kana, I couldn't look up translations for most of what I didn't know. Super discouraging.
に particle is an indicator of a specific time. When you say 今週の日曜日(konshū no nichi youbi), it translates to "Sunday this weekend" which is already specific enough that it does not need the particle. However, if you arr talking specifically about just Sunday, then you need to add the particle に.
The common link between weekday names in Romance, Germanic, and Japanese (through Chinese) weekday names seems to be the seven celestial objects visible to the naked eye: the sun, moon, and five planets. For the planets, Latin used the names of Roman gods associated with each planet; German does likewise with Norse gods. Japan uses the five elements and their associated planets. See https://en.m.wikipedia.org/wiki/Names_of_the_days_of_the_week
The days of the week have essentially three parts to them - the first part tells us the name of that day - getsu, ka, sui, moku, kin, do and nichi. The middle part you 曜 tells us it is a day of the week. And the last part tells us it is a day 日。So 日曜日 - 日 nichi - tells us the name of that day or helps us know which of the days it is. 曜 you lets us know that it's a day of the week, I guess as opposed to a holiday 祭日 さいじつ (saijitsu) or tomorrow 明日. And 日 hi/bi reiterates again that it is a day. So nichiyoubi - Sunday. And as everyone has already mentioned - there's multiple ways to read each kanji.
I wonder why do we need the 今週の? If we say Sunday isn't it obvious? Also why don't they use the ましょう ending to convey "shall we", because sounds like a question - not an invitation? Finally, where is the "go" translated part of the sentence, for example, using 行って before the final verb.