Re rent: under an assured shorthold tenancy, if the tenant is still present when the fixed term ends, the tenancy automatically becomes a statutory periodic tenancy. The tenant must still pay the rent, and the landlord is entitled to keep it.
None of this has any bearing on a Section 21 notice, as the judge's role is purely to determine whether or not the landlord has followed the correct procedure.
Re time: I don't have personal experience of this; but from reading around the subject, it seems that we're talking weeks rather than months. This may be because the tenant will already have h…