No, not necessarily. To work as a PD, you must be a designer:
You must have the right mix of skills, knowledge, experience and (if an organisation) organisational capability to allow you to carry out all of the functions and responsibilities of a PD for the project in hand, and be in control of the pre-construction phase.
The client has the duty to appoint the PD. If you are no longer involved with the project, the client must appoint another PD. You will still be responsible for the design decisions you made in preparing the original plans. The new PD must then take responsibility for any design decisions made following their appointment.
If the client fails to appoint the PD, then the client assumes the role.
Yes. You can choose to take on the PD role yourself. If you do you must have the skills, knowledge, experience and (if an organisation) organisational capability, to carry out all the functions and responsibilities of the PD.