This: (https://web.ku.edu/~edit/that.html) might be useful in explaining why 'that' can be omitted - there are some good examples there in my opinion. As far as I know, "that" can't be used as a relative adverb; it's a relative pronoun. Using the word 'which' makes some of these sentences more complicated than they need to be, and may be confusing. I'll just pick out the ones you have mentioned: "This is the place that I used to live in." This is fine, however the final sentence there sounds more natural: "This is the place where I used to live." I would personally say: "This is where I used to live.", or "I used to live here." "She died on the day that he arrived on." This is incorrect, as you've already used the preposition early in the sentence, so there's no need to place another at the end. "She died on the day that he arrived." This one is the most natural. "She died on the day when he arrived." This one is incorrect, as 'on the day' and 'when' both define a time (with 'on the day' being more specific). You can only use one of them. "She died when he arrived."/"She died on the day he arrived." Again, you could remove the preposition entirely ("She died the day he arrived."), but it sounds fine either way. "This is the reason that they were defeated for." This one is incorrect. With 'reason for', there is a noun involved. For instance "This is the reason for their defeat." In any case, with 'reason', these are usually omitted. The most common sentence would be "This is the reason they were defeated.". "I don't know the way that he solved the equation in." I can't describe the grammatical reason for this being incorrect, as I'm not entirely sure. 99% of the time, the word 'how' would be used instead of 'the way that', like in your final example. These are the most natural sentences that occur to me: "This is where I used to live." "She died on the day he arrived." "This is the reason they were defeated." "I don't know how he solved the equation."

Most of the sentences using WHICH don't seem natural. The most natural way to say these are: This is the place where I used to live. She died on the day he arrived. This is the reason they were defeated. (or This is why they were defeated). I don't know how he solved the equation.