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1. No, simply receiving mineral rights does not mean "doing business in" a particular state, but since they are for the rights of property in Ohio, Ohio tax returns and tax apply to that income. So, technically no but the result is the same as far as being subject to tax in Ohio.
2. How the check is written out and where the funds are deposited does not matter. The result is the same. The Corp has Ohio income. The trustee is not a "pass through" he is an agent of the Corp. So, the checks can be written to him or the corp but will be reported as income to the Corp. regardless.
3. Where the bank is makes no difference.
4. No entity can hold the income tax free. "Pass through" means only the owners pay tax not the business entity, and you have it backwards. In a pass through structure, the owners are taxed on income whether distributed to them or not.
The best thing to do would be to make a Sub S election for the C corp to eliminate Corporate tax if the Corp qualifies to do so. Otherwise, there is not much that can be done to change the tax situation.
I understand what you are asking but no such business or trust structure exists by which royalties or other income can be received but not taxed until distribution to owners. A C corp is the only thing close taxing income at the corporate level (generally the net tax is less) and then not taxed individually until dividends are made.
If the only income is from these rights and the Corp qualifies, a Sub S tax election would be more fitting.