Cheating?

Would it be considered cheating if I play correspondence game, while having a board at home where I play along and try different variations for next move, instead of doing it just in my head?

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In my opinion it is not cheating because you don't get any help from the 3rd side. If it is cheating, then nearly all the chess grand-masters from the past could be considered as cheaters and they were also using help form the other people to analyse their ongoing games. So, you can feel yourself safe in that question :)

For me very interesting is the next situation in pro shogi:
When pro players play 2 days game and the first day finishs - can they use engines or 3rd side help to analyse their game before the start of the 2nd day?

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You don't need a physical board to calculate moves. Correspondence games have an option called "Analysis board" (the microscope icon).

From Lishogi ToS:

"1. Cheating. We define this as using any external assistance to strengthen your knowledge and, or, calculation ability to gain an unfair advantage over your opponent. Some examples would include computer engine assistance, opening books (except for correspondence games), endgame tablebases, and asking another player for help, although these aren’t the only things we would consider cheating."

It's not cheating because you use your own brain and not an engine (external assistance), and there's no unfair advantage (your opponent can use the same tool).

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I wasn't aware of this function, I felt like I was trying to artificially extend my calculation where my mind cannot go, but now I understand it is expected and fair thing to do.

In pro 2 day games I guess allowing analysis after first day is absurd, but how do You expect players to refrain?
Sealed move doesn't solve the problem, there are probably just a few moves that You expect Your opponent to play.

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In correspondence, this is allowed to my knowledge... however it is not allowed in live games. In correspondence you are even allowed to use an opening book.

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If memory serves me correctly, when the 2-day title matches are played, they take place in the JSA headquarters; and since an incident in 2016, outside technology is forbidden during the match and the players can't leave.

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