1. What types of skills do managers need? 2. Does the importance of these skills change depending on managerial level?

There is a consensus among management experts that the types of skills managers need include critical thinking, financial and accounting fundamentals, communicating and responding, conflict resolution, coaching in job performance and motivation and corporate vision and values, strategic thinking and planning, group collaboration, and project management skills.
Critical thinking skills allow for analysis of the established ways of doing things and of perceiving things. Critical thinkers ask "Why?" and "How?" things are, have come to be and should remain or be changed. They also ask "What?," for example, as in "What would a new hire for my position do to fix the problem(s) we're facing?" Critical thinkers seek the underlying factors and the most effective directions through critical analysis of available facts and through gaining perspective by seeing through others' points of view.
Financial and accounting skills provide the means for casting accurate budgets, interpreting financial reports, forecasting return on investment reports, and projecting costs and profits for projects.
Communicating vital information is a paramount responsibility for managers who are tasked with downward communication of knowledge covering everything from job descriptions, task processes, projects specifications, production design or changes, to corporate vision, organizational values and ethics and corporate structural changes.
Conflict resolution is of growing importance as global corporations include an ever increasing array of individuals from diverse backgrounds and varied cultural and ethnic groups. 
Coaching is valuable as a tool to increase and direct job skills and motivation and to bring employees into alignment with the corporate vision and values so that vision, values and ethical behavior are homogeneous horizontally and vertically throughout the organization.
Strategic thinking and planning are requisite for meeting corporate, department and project objectives and goals.
Collaboration in teams and groups that is effective builds trust throughout and between groups and teams. It breaks down barriers to efficiency produced by divisive informal separation. Effective collaboration engenders optimal performance and maximizes outcomes and outputs.
Project management facilitates successfully led, directed and motivated projects from beginning to end so that all participants work according to the same project language, with the same project tools and toward the same outputs or outcomes.
https://www.poynter.org/reporting-editing/2012/10-key-skills-todays-leaders-need-to-succeed-in-2013/