Does the debt load affect the volatility of equity?

Q1 - Yes, debt load has an impact on the stock price. For instance, say you are valuing a company with a discounted cash flow model, while the interest won't affect the operational cash flows, it will increase the cost of capital. With that, the perceived value will be less than a similar company with less debt. Debt will also affect the volatility of the equity. As debt becomes a larger portion of the old Assets = Liabilities + Equity equation, changes in asset value will have a larger sway on Equity.

Q2 - Depends. If the market perceives that the company needs more debt to fund capital expenditures due to opportunity, then the market should react positively. If they're issuing debt and the market thinks this is a poor choice, the market with punish the stock. As a caveat, if they issue debt to buy back stock the market will almost always act positively due to less shares for trade