You have asked many questions.
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If the tax paid is recoverable, apply the proper tax code to the expense claim just as you would if purchasing from a supplier. Of course, you should keep all the documentation to justify your claim.
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The expense claim will show the tax service that the company reimbursed someone for a company expense. The fact that the payment went to an individual is not conceptually different from a payment going to a credit card processor if the company had purchased directly with a credit card. In neither case did the payment go to the merchant who sold the printer.
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Capital accounts represent the investment interest in a company by its owners or partners. They reflect a form of equity. They can be used for single or multiple owners. But they are required for shared ownership (partnerships).
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Owner’s equity is the term normally used to represent the investment interest of a sole trader or proprietor. In this form of organization, the proprietor owns the entire company, so retained earnings, capital accounts, and in fact every other aspect of the company is owned by the proprietor, including liabilities. Many tax authorities consider a sole proprietorship to be simply an extension of personal finances. In these situations, companies do not file separate tax returns, but proprietors report company results as part of their individual returns. So recording owner’s equity is just a simpler way of handling accounts. See this Guide:
https://forum.manager.io/t/simple-equity-accounting-for-sole-traders-proprietors/6971