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I have been working on implementing Jozsa-Deutsch decision algorithm, Grover search algorithm, Shor factorization algorithm on my home-made 2 qubits device.
I am wondering if there are any other breakthrough quantum algorithms that I can devise, run and experiment on with on my 2 qubits device?
Probably only two qubits constrain set of algorithm you can run on your device, for example HHL algorithm (linear equation solver) seems to impossible work there.
You can however implement algorithms based on uniformly controlled rotation used for preparing arbitrary quantum state. See details in the paper Transformation of quantum states using uniformly controlled rotations.
Also you can implement toy-model (adding two single bit numbers) of adder based on Fourier transformation. See paper Addition on a Quantum Computer.
Moreover, you can play with quantum tomography on one and two qubits. See this thread for more information how to do the tomography: Implementation of tomography on IBM Q.
I have implemented via qiskit two algorithms and posted in github that need only 1-2 qubits. First is Iterative Quantum Phase Estimation (IQPE) that works on two qubits, the second one is Variational Quantum Eigensolver (VQE) that works on only one qubit (one can do also for 2 qubits) in my implementations. Actually they are jupyter notebook tutorials, so I hope it will be easy to understand what I did. Here is the link to the repo where you can find both of them:
Stephan Jordan maintains a very comprehensive list of quantum algorithms on https://quantumalgorithmzoo.org/. Now that you have worked on some famous ones, you can consider veering off a bit. Take a look at the "Algorithm: Machine Learning" consolidation section. There are for instance single-qubit and two-qubit classifiers that could be interesting. In particular, the "Quantum enhanced feature space" classifier that was published in nature last year can be done on 2 qubits (https://arxiv.org/abs/1804.11326).
This is not an full answer to your question. But I am very interested in how did you build your own 2-qubit device, as I am thinking about such a project for some months now. Would you mind to get in touch with me and exchange our thoughts on this? I would really love to get into discussions with you, maybe you could support me to build my own device?
Regarding your questions, there are a lot of tasks in the MS Quantum Kata series, that only work with one or two qubits, maybe you'll like some of them, e.g.
https://github.com/microsoft/QuantumKatas/tree/master/CHSHGame
https://github.com/microsoft/QuantumKatas/tree/master/Superposition - Hardy-State
There is
A Polynomial Quantum Algorithm for Approximating the Jones Polynomial
For simple knots, you can do that with 2 qubits...
