EnviroTech plc is an international company in the UK$.$ The company is dedicated to protecting and preserving the environment through innovative and sustainable solutions. Its aim is to minimize the negative impact of human activities on the environment, promote conservation, and create a sustainable future for generations to come. By combining expertise, innovation, and a passion for the environment, EnviroTech plc strives to be a leading force in environmental management, creating a sustainable future for all. The company is financed by both equity and debt. The debt to equity ratio of EnviroTech plc is $1.$ The book value of total assets of the company accounts for $9,000$ million.

The debt is in the form of long$-$term bonds, with a coupon rate of $12\%.$ The bonds are currently rated AA and are selling at a yield of $13\mathrm{.}50\%.$ The market value of the bonds is $85\%$ of the book value. The firm currently has $45$ million shares outstanding, and the current market price is $75$ per share. The firm pays a dividend of $4\mathrm{.}5$ per share and has a price$/$earnings ratio of $12.$ The stock currently has a beta of $1\mathrm{.}45.$ The six$-$month Treasury bill rate is $6\mathrm{.}5\%.$ The market risk premium is $5\mathrm{.}0\%.$ The tax rate for this company is $25\%.$

EnviroTech plc is evaluating its capital structure and considering a major change in its capital structure. It has the following three options:

Option $1$: Issue $1\mathrm{.}4$ billion in new stock and repurchase half of its outstanding debt. This will make it an AAA rated company. AAA rated debt is yielding $10\mathrm{.}5\%$ in the market place.

Option $2$: Issue $1\mathrm{.}5$ billion in new debt and buy back stock. This will drop its rating to A$-.$ A$-$rated debt is yielding $14\mathrm{.}25\%$ in the market place.

Option $3$: Issue $2\mathrm{.}5$ billion in new debt and buy back stock. This will drop its rating to CCC$.$ CCC rated debt is yielding $18\mathrm{.}35\%$ in the market place.

Calculate the cost of equity, after$-$tax cost of debt and the cost of capital under each option. From a cost of capital standpoint, discuss which of the three options the company should pick or whether the company should maintain its current capital structure.

I have seen the solution provided here and further have a few questions as below:

$1)$ If beta value has already been used to in calculation of equity, is it necessary to use it again in calculation of WAC?

$2)$ In Option $1,$ why is $0\mathrm{.}5$ billion deducted from the debt?

$3)$ How do we assume the beta in each options? Can you further explain levered and unlevered beta?