Form and effect of the intrinsic Hamiltonian on the IBM machines

I want to understand the form of the intrinsic Hamiltonian of the IBM machines, such as ibmq_16_melbourne, and how this affects the data I get out when waiting using identity gates.

From the ZZ-characterisation tutorial in the documentation, it seems like the general form of the intrinsic Hamiltonian is
$$
mathcal{H} = sum_i^N frac{2pinu_i}{2} (1-sigma_i^z)+sum_{ij}^Nfrac{xi_{ij}}{4}left(1-sigma_i^zright)left(1-sigma_j^zright),$$
with sums in theory over all $N$ qubits. Here $nu_iapprox 5,text{GHz}$ and $xi_{ij}$ seem to take values up to $sim 100,text{kHz}$, but only significantly above $0$ for nearest neighbours. Is this correctly understood?

Now, if I prepare an initial state $frac{1}{sqrt2}left(|0rangle +|1rangleright)otimes|0rangle^{N-1}$ I do not see oscillations with a frequency anywhere near $5,text{GHz}$, but rather something closer to $100,text{kHz}$. Does this mean that the measurements are being taken in the co-moving frame with frequencies for each qubit being $nu_i$? If so, why do I have these residual oscillations?