How is control and consolidated interest determined with indirect shareholdings (e.g., parent → subsidiary → sub-subsidiary)?

What is the key accounting distinction between an ownership change that retains control and one that loses control?

When the parent's ownership in a subsidiary decreases (control retained), what happens to the acquisition differential?

Before obtaining control, how is a small equity investment that is later stepped up to control reported, and what happens at the step-up?

How does the parent account for selling some of its subsidiary shares while retaining control?

How are a subsidiary's preferred shares considered when allocating consolidated net income?

What happens to the parent's interest when a subsidiary issues additional shares to the public?