this is possible but would require a custom Approval process (for the selection of the Approver) and an Approval Chain. You could also select based on the answer to a question e.g. Cost Centre and auto generate the approver so the end user does not have to pick.
Notifications can be used (along with Alternate Approvers) to remind of time frames. Manager Approval does not automatically send up to the next Approver, this requires an approval to achieve and multiple levels enabled in the Manager process.
Alternate Approvers can be used to delegate the Approval when a person is unavailable e.g. On Holiday, Sick, etc.
So all in all your requirements are achievable OOB with configuration, this can be further extended using Process Designer if you wanted to go down this road.