If there's missing rent based off the 10% I'd request a breakdown of everything. Usually, this would be in the form of a rent roll and contains everything related to your property related to vacancy, rent collected, expenses, fees, etc.
Since I'll be taking over in the middle of the tenant’s lease, what actions do I need to take with the previous property manager?
Depending on your situation, you may want to consider having the tenants sign a new lease. Before taking any action, make sure to request all pertinent info on your existing tenants. The property manager should provide a copy of the existing lease agreement, rent roll, other related documents, and notify the tenants of a change in management.
You should also be aware of any specifics outlined in the lease for your tenant's situation (e.g., pets).
How should the contract be handled moving forward?
You will want to consider having the tenants sign a new lease agreement if you believe that's necessary based on the existing lease agreement.
If you're referring to your agreement with the property manager, make sure there's no additional costs or hidden fees.
What is the best way to officially inform the tenant about the management change?
Phone call with email follow up or email so there's a paper trail. You'll also want to make sure the tenants know where they will be paying or sending rent to moving forward once the change has been made.
Do I need to draft a new lease with the tenant, or can I continue with the existing one?
You can continue with your existing one, but may want to consider having a new one drafted.
How should the security deposit be managed in this transition?
The property manager should transfer those funds to you. Those funds should be held in a separate account.
What specific responsibilities does the previous manager have to ensure a smooth handover?
They should notify the existing tenants and let the know to expect contact to be made by you.
Lastly, would it be better to introduce myself to the tenant as the owner, or as the new property manager?
As the owner or landlord you may be able to build a more personal connection with the tenants. In some cases, anonymity is healthy, but this depends on your preference.