APV, the ungeared cost of equity and asset beta

I am not sure about some assumptions I made on APV and cost of equity. I would be very greatful if you found the time to answer my questions.

If I use an APV approach to evaluate a project, I would value the project cash flows as if they were all equity financed, right?

Had I to use the Modigliani Miller Formular, I would use the ungeared cost of capital (kei) to discount the Present value of the project values, right?

Furthermore, if it came to ungearing beta values in an APV approach in the exam: Is it appropriate to calculate and use the asset beta (the ungeared beta) to establish the discount rate? So would I use the asset beta in my CAPM formular to come to the right cost of equity?

Furthermore, independently from what valuation method I use (APV; NPV…) if I decide to finance a project entirely from equity, would I use an ungeared beta when calculating the cost of equity with the CAPM formular?

Thank you.