Is it permissible for the old tenant to take "key money" from the new tenant in exchange for vacating the apartment, knowing that the landlord has agreed to this and to increasing the rent for the new tenant?

The 'key money' (khalaw rujul) has permissible and impermissible forms. Among the permissible forms is when the agreement is made between the old tenant and the landlord, or between the old tenant and the new tenant, during the lease period. Since your lease period has not yet ended, there is no objection to you taking key money from the new occupant. The landlord does not have the right to raise the rent as long as your lease agreement is valid.

As for the security deposit that the landlord takes from the tenant: He should only take from it what the tenant has damaged through transgression or negligence. The landlord is not allowed to use this money during the lease period; otherwise, it would be a loan, and it is impermissible to combine a loan with a lease (salf wa ijarah).