Given a simple quantum system, how do I derive its Hamiltonian?
For quantum systems of continuous variables, the most common way to construct the Hamiltonian is to add the kinetic energy and potential energy, as described in this resource. The kinetic energy part is explained here, and various potential energy models are given here:
which unfortunately I could only get to from this page!
For quantum systems on discrete variables, you can construct any $2^n \times 2^n$ Hamiltonian using the Pauli matrices, and any Hamiltonian of any dimension using generalizations of the Gell-Mann matrices.
In terms of other resources:
There is an open source Hamiltonian Zoo on GitHub, but it is very incomplete. So far it tells you how to derive the Hamiltonian for two charges interacting with each other (Coulomb), for a spin system interacting with a magnetic field (Zeeman), and for a 2D p-wave Fermi superfluid, but not much else. However, since this is a resource request, I think the Hamiltonian Zoo is a good starting point, because it lists the names of almost every mainstream Hamiltonian imaginable, and the best resource for learning about each of those listed Hamiltonians is Wikipedia. For example:
In every case, the Hamiltonian is in the article, just look for the equation containing the big $H$!
Given a Hamiltonian, what questions can I answer about the system it describes (and how?).
There is no "resource" I know that teaches people what can be learned about a system based on looking at its Hamiltonian, and I'd be quite surprised if such a resource existed. What I can tell you is that there are things about the system that can be learned by looking at the Hamiltonian (such as number of particles or number of degrees of freedom, by looking at the number of terms in the kinetic and potential energy operators in the case of continuous variables, or the size of the matrix in discrete variable Hamiltonians). However you may want to ask this part as a separate question (and not a resource request) in case other people want to suggest other things that can be learned about a system by looking at the Hamiltonian apart from what I've already told you.