Chemistry - How were x-ray diffraction patterns deciphered before computers?
I agree with @andselisk that this question is quite broad. I will focus on two specific questions asked
The only equation I know for x-ray diffraction is Bragg's Law but is this the only equation used to interpret the data? [...] How do you translate the spots on a detector to electron density plots using Braggs law?
Apart from Bragg's law (which tells you where the diffraction spots are for a known orientation of a crystal with know unit cell), it was also known that real space (electron density) and reciprocal space (diffraction pattern) are related by 3D Fourier transform. For structures containing one or two atoms, just knowing the unit cell parameters and the symmetry is enough to get the entire structure. For anything slightly more complicated, the Fourier transform of amplitudes or later, of intensities (Patterson methods, http://reference.iucr.org/dictionary/Patterson_methods), had to be used. The first crystal structure analyses were of crystals with centrosymmetry, where the phase problem is easier to solve.
My question is what is the general process for doing this without computer software?
Diffraction data was measured on film, with gray-scales to assess intensity of signals. To calculate a Fourier transform, pre-computed tables were used, such as the Beevers-Lipson strips. As Andselisk commented, Fourier transform was used late in the 20s, and initially for problems that were one- or two-dimensional. There is a nice account by P.P. Ewald available here: https://www.iucr.org/publ/50yearsofxraydiffraction/full-text/structure-analysis This was written at a time when the myoglobin structure had just been solved, after "50 years of X-ray diffraction".
Not just in the '20s but up to the '90s at least, the d-spacings were estimated by hand measurements of diffractometer peaks or film lines and applying the Bragg formula. d-spacings were then ranked from the most intense down to the least intense. Starting with the 2 most intense d-spacing values - and allowing +/- 0.02 angstrom error margin to each - a process of searching the Hanawalt Search Manuals was done till a chemically plausible phase with these two values was found. If no plausible phase was found for these two most intense d-spacings, the second value was replaced with each of the third/fourth/fifth/etc most intense d-spacing value and another search made till a chemically plausible phase was found. Once located, the remaining less intense d-spacings listed for this phase were compared with the remaining calculated d-spacings and if found this was supporting evidence for this phase. Remaining d-spacings were relisted according to intensity and the process repeated. It always seemed daunting at first but, with a little initial guidance from a more experienced person (usually another postgrad student) and with a shortlist of all plausible phases plus their d-spacings beside you, it got easy enough in metallurgy and ceramics. It must have been much harder in biochemistry but I guess those working in that métier had their own shortcuts.
For more detailed work, e.g. for residual stress calcs, grain texture effects, part-crystalline/part-amorphous phases, exploration of short-range ordering in amorphous phases, etc one referred to texts by Cullity and Azaroff. Beyond that you had to look at papers by other researchers, talk to x-ray lab technicians and above all try to look at the problem in the way that was most natural to oneself or most amenable to the nature of the problem.