1. The Product Owner is not a mandatory attendee at the Daily Scrum, which is only for the Development Team. If the Product Owner does turn up, they should not speak or take any active role in the meeting.
2. A Product Owner reflects the wishes of the stakeholders and is the single decision maker in terms of ordering items on the product backlog. They can appoint a proxy but it's considered a poor practice as you've added a layer of communication and increased the chances that the actual Product Owner might overrule a proxy's decision(s), which negates some of the positive elements of scrum. This could be a huge waste and very harmful to productivity.
3. There is no role called Project Manager in scrum and best advice is to avoid appointing proxies, where possible.
4. As point 2 above. The Product Owner reflects the wishes of the stakeholders. They are a central point of reference for both stakeholders and the scrum team.