I have a couple questions regarding khums.
1) If I have for instance, shoes or clothing or rings that I have not worn in a year, is khums mandatory, and if so, what is the calculation?
2) If I have a personal savings that I have for my children who are underage, so they are financially stable when I am no longer alive, is khums mandatory on such a savings for underage children?

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1- Yes if you bought those items not from money that khums was already paid on, and one year passed on them without you using them, you will pay khums on them. On the day of paying your khums, estimate their current market value (not their value when you bought them new), and then pay 20% of their value as khums. Current market value means how much are you able to sell them today if you wanted to.

2- The fatwa of your Marja’ (Ayatullah Khamenei) is that khums is not mandatory on money owned by children. If you gift this money to them, they own it. But if it’s yours (you have not gifted to them and you are the owner) then yes you’d have to pay khums on the savings.