Basic Series/Parallel connection understanding
Have I made any mistakes in re-drawing my network?

Yes. You left R2 out of the re-drawn network entirely.

Is there any foolproof way of telling which connections are in parallel and which in series in any given network?

If two components are end-to-end so that all the current flowing out of one must flow into the other, then they are in series.
If two components are connected at both ends, so that the exact same voltage must appear across both components then they are in parallel.
Sometimes you will find that neither situation applies, and then you will have to use more complicated analysis methods that you will probably learn shortly.