No, don't use "in" here. It should be "on". Nobody says "in this day". Everyone says "on this day". Exception being "in that 'day & age' ", but that kind of expression is not relevant here.

no, you can't. Because "day" always use "on" preposition. eg. on my birthday, on the same day.

Yes, you can say "in the same day". "In" might even be the superior choice because it emphasizes the time span. "On" emphasizes a date, like Monday. But the short time span is the amazing thing. "In" is the preposition used to describe containment. The day contained different types of weather, so "in" is the best choice. Rely on logic to make such choices, not on what people or google might tell you is most commonly said.

In the US, only ‘on’ is correct in the given sentence. You could possibly replace ‘on the same day’ with something like ‘within the span of a single day’ but it doesn’t add anything. You could also use ‘in’ with a different meaning of ‘day’ where you are referring to ‘daytime’. ‘Daytime’ and ‘nighttime’ are the portions of a day when the sun is up and down, respectively. For example, ‘In (the) daytime, temperatures are pleasant, but be prepared for cold nights.’ (Note that in my example, it is possible to omit the ‘the’ before daytime. Doing so gives the sentence a slightly more general sense.) Replacing ‘daytime’ with ‘day’ in my example is not a good or precise way of speaking, but it is something that you could probably hear native speakers say. From the context, this is not the meaning of ‘day’ in your example, so only ‘on’ works in your example.

'On the same day' is correct. A good way to check is to type both phrases into Google (or your preferred search engine) in inverted commas. If you Google "in the same day", the only results are questions similar to yours. If you Google "on the same day", the results are such questions and also real texts containing the phrase - articles, blog posts, etc.