About gearing, Beta, WACC and cost of equity

Can I say that when there is no gearing (all equity financing), Beta assets = Beta company = Beta equity? While WACC = cost of equity only?

When gearing is added, the above Beta assets or Beta company can then be re-calculated to a new Beta equity, this new Beta equity can be used to find out a new Cost of Equity through the CAPM formula?

If the gearing or capital structure has undergone a big change, Adjusted Present Value is much better than a new Beta or a new WACC?