1. Yes, it is perfectly permissible to purchase a property in a personal name and to have a company of which you are shareholder pay you rent. The law does not prevent this occurring. However, you should ensure the rent is the open market rent and does not exceed the market rent, as otherwise you could run into trouble if you paid yourself an excessive rent. However, it is lawful to have a limited company of which you are shareholder pay you rent for a property rented. The company should occupy or lease the property. Here, I would recommend you draw up a written lease to document the transaction.
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3. The open market rent is an issue because there are anti-avoidance measures for companies which prevent the amount of the rent which is deductible in law from company profits exceeding what is the open market rent. There is no difficulty with a management company charging appropriate fees. However, again, there are anti-avoidance measures which prevent the level of deductible expenses exceeding what would be available if the services were contracted for, in the open market. The idea behind the legislation is that there should not be abusive deductions from company profits which exceed what the market would charge.
4. You can certainly use the company as the managing agent. There is nothing to prevent this and you can earn a proportion of the monies charged in rent.