Shandy
Yes, you have to claim the State Pension, you should receive information about this about three months before you reach State Retirement Age.
Your ESA will stop when you reach SRA.
Your PIP will continue past SRA as long as you continue to meet the criteria for an award, the only limitation is that you cannot receive a new award of the Mobility component or an increased one once you reach SRA.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems