Hi Nicola,
My original post here, if you click ‘Show Full Post’, explains how increasing rent works in England.
Essentially, your property manager seems to be wrong on the first count; i.e., a tenancy renewal is a normal time to propose a rent increase, since it is a new agreement between parties and forms a new tenancy.
On the second point, you can ask a tenant to leave, but they are under no obligation to engage with that request unless you serve notice. You would need to give three months’ notice, not 60 days.
Sam