Let us look at each observation and question in perspective. Before delving deep into the questions, please let me share a few reference architecture diagrams on the components of a quantum computer.
We need to review the mentioned observations and understandings from a practical implementation vantage point. When we consider the quantum realm in its entirety, there is a finite and fundamental thermodynamic limit between the forces and fields of quantum physics and classical physics.
We cannot say that a quantum computer is at the highest level of abstraction controlled by a classical computer. A quantum computer is a physical realisation of a quantum Turing machine supported by quantum mechanical processes which are modelled as physical qubits and abstracted as logical qubits.
There is an interface between quantum mechanical processes and classical computer processes. Through this interface, input data from a a classical computing device can be fed into a quantum circuit. This quantum circuit can help us to realise a set of quantum gates which helps us to create a quantum register necessary for quantum circuitry.
Quantum Circuits are constructed from Quantum Registers which are devised from various types of universal, special purpose and custom built qubits. The logical quibits can create different permutations and combinations of physical qubit manifestations.
Quantum register is a type of circuit construction from logical qubits. Perhaps we will devise more diverse forms of circuits such as quantum stack machines, quantum memories etc. in the near future. The following diagram depicts how quantum circuits function in the physical circuitry.
Would the hardware structure of a quantum computer, that I asserted above, be correct?
There are various hardware models and architectures for quantum computers. One of the early model is based on Nuclear Magnetic Resonance. Adiabatic Optimisation process based Quantum Annealing Computer is built by DWave. Superconducting Qubit based Universal Quantum Computer is the model implemented by IBM. Intel is implementing Silicon based Quantum Computers. China is building Photonics based Quantum Computers. Microsoft is pioneering a topological quantum computing model.
We cannot say all of them follows the same exact hardware components and models for a quantum computer. However IBM has put forward an interesting model for the super conducting chip based quantum computing model. Please find it below.
And how is a classically written quantum code, such as Qiskit, able to instruct the quantum computer to run a certain quantum algorithm (where the algorithm is simply comprised of quantum logic 'gates' that cleverly manipulate the qubits)?
I believe you must have got a clear idea of how a Quantum Algorithm can be implemented in a Quantum Programming Paradigm like QIS Kit from IBM and PyQuil like a language by Rigetti or Q# by Microsoft by seeing the first architecture diagram. Every general purpose Quantum Computer has been implemented wit a Quantum Compiler + Quantum Runtime and a Quantum Arithmetic Unit and a Quantum Memory in the conventions of classical Von Neuman Computer Architecture. They will be supported by Quantum Processor with specific instruction sets and Quantum error correcting codes as part of the instrumentation layer. They will be supported by the Quantum Processors and Quantum Chips. If you connect to IBM Q on IBM Bluemix you can see the topology of quantum processors for various quantum computers.
Hope I could address some of your questions. Looking forward to further discussions.