SPOT uses SI units throughout. The "Amplitude" of a quadrupole is given by gradient x charge/momentum, so units are m^-2. See Eq. (5) in our PRST-AB 064202 paper.

All quantities  related to the K-V envelope equations refer to "effective" quantities for the equivalent beam: envelope radius means 2xrms radius, emittance means 4xrms unnormalized (sometimes also called "edge" emittance). Chapter 5 in Martin Reiser's book "Theory and Design of Charged Particle Beams" has a good review of this.

(Notice that SPOT cannot use Twiss parameters directly, unlike matrix codes like TRACE.)

When you say "reference trajectory" you probably mean the desired "average" behavior of the envelope, as explained in the Appendix of our paper on matching. The "ref. trajectory" is not necessary for the calculation of periodic envelopes like in Figure 5 of the paper. Instead, it is used in problems like the one illustrated in Figure 9. To calculate periodic envelopes, you run "Matcher", and click "solution" to refresh the envelope plots when the calculation is done (which is fast.) To help you understand this, I'm sending you two SPOT files (extension "spi") that correspond to Fig. 5 (100 mA, 10 keV  electron beam). The beam is already matched in both cases (one with smooth quads, the other with equivalent hardedge quads), so try to mismatch the beam by changing the initial beam size in one plane, and run "Matcher" to match it again.

Normally, you run "Matcher" first to find your periodic target envelope parameters (this would involve only two quads in UMER, as in Fig. 5) and then, with the full matching  beamline
(which normally involves 3 or 4 quadrupoles at least) you run "Local Optimizer". I will send you an example file of the latter in another e-mail. Try the simpler "Matcher" first. Also, review Section V of the paper before trying the "Local Optimizer". I'll be glad to assist you with any other questions.