Paul Jerram, PMP
I'll respond to the first two questions only, as I work for Management Concepts, a premier provider of training and professional services to the federal government, located in Tysons Corner, VA, and not in the public sector.
Question 1: many of us know, from firsthand experience, the term "The Accidental Project Manager", and I definitely, in years past (pre-certification and experience but already with the "scars"), qualified for that title. Also, many people routinely handle the duties of a project manager without necessarily having the formal education or the PMP credential, for example. It also really helps for project team members to have a good grasp of what "project management is all about". We at Management Concepts designed a PM training product around that very topic and belief; "Project Management Essentials for Team Members". Check it out at: https://www.managementconcepts.com/portal/server.pt/community/training/301/course_detail?mcTarget=course&mcTargetID=618112
Reading down some of the other contributions here, I also completely agree that "practice makes perfect" (or at least, better and more confident). The more you focus on the discipline, use the techniques and learn from your mistakes, the greater the chance that you will improve. It does get to the point though when it's beneficial to ask yourself the question: is it time for me to take advantage of some of the certification opportunities that are out there? They can demonstrate your capabilities in the skill, and in the right setting, cause your organization to have confidence in your capabilities, as evidenced by your certification and continued contributions to projects.
Question 2: Project or Program Management Offices can have a great role to play in organizations on many levels. However, I think there's a case to be made to instill greater numbers of people with project management skills, as more people become engaged in the process and more readily understand how they fit into things and how they can become key individual contributors.