For point 1) you are correct no instrument can be precisely calibrated or has infinite precision. This is in part limited by how well the corresponding SI units are known NPL has a nice little faq on this. Similarly all measurements will have some noise (possibly very small) which limit precision.
Personally I wouldn't use weight as an example. There are several reasons. Firstly as you point out it is easy to confuse ideas of mass and weight. If you want to be completely correct this is a confusion you don't need. Another concern is that mass is currently the only fundamental unit that is still defined by a physical artifact (the standard kilogram) rather than by physical constants so the definition of a kilogram is pretty uncertain.
In my opinion a better example would be measuring a metre. A metre is defined as "the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second.". Now the question becomes how good is your clock. Now I'm going to use an atomic clock which is accurate to $1$ in $10^{14}$, but this clearly has some small uncertainty. In reality you probably don't use something this accurate but something which has been calibrated against something that's calibrated like this and so will be much less accurate.
As a side note this is not how a metre is actually calibrated, people actually use things like a frequency stabilised laser, which has a very well known wavelength, and an interferometer to count the fringes seen over a distance.
For 2) I don't think you need to say any more. There are lots of things you could say but you are trying to answer a specific question, not write a book. The NPL:begginers guide to uncertainty of measurement provides a good introduction to some of the topics, but iss by no means comprehensive.
For 3) I would say your analogy isn't far wrong. Its only really scientists that care about this sort of accuracy. Possibly also anyone involved in micro-manufacture, think Intel. Even most engineers don't care (they tend to double stuff just to be certain ;) ). I think the best way to show it is to do hat you did in your actual answer and give this as a percentage error to show how small it actually is.
Experimental results can deviate from ideal. Outcomes depend sensitively on small differences in mass and alignment, and the extent to which kinetic energy is conserved. Alignment of non-identical balls is much more difficult. Considerable effort and expense may be required to achieve reliable results.
Although Simanek displays his apparatus, his webpage mostly contains animations. The only video clip (ooO <--> Ooo) does agree reasonably well with your simulation (#2 and #5). Perhaps his other videos were less convincing and left out for that reason.
Simanek's observation that the OoO collision is symmetrical might have been correct for the size of balls he used. (Judging by his video clip, I suspect that his apparatus did not actually show perfect symmetry.) In your simulation, although the 2:1 mass collision is not symmetrical, the 4:1 mass collision is much closer. Probably the n:1 collision gets more symmetrical as as n --> infinity.
Your simulation implements conservation of kinetic energy as well as momentum perfectly, so it ought to be more reliable than experimental results, which cannot guarantee perfectly elastic collisions and perfect alignment. However, investigating the differences with experiment may unearth interesting physics, particularly when the "successive impact model" does not apply.
The following Wolfram simulation includes the transfer of compression waves along the chain of balls :
http://demonstrations.wolfram.com/PhenomenologicalApproximationToNewtonsCradle/#
There are a number of very detailed and informative answers on the operation of and ideal conditions for Newton's Cradle already on Physics SE, such as
Newton's cradle
Newton's Cradle: why does it stay symmetric?
The group at California Institute of Technology (cited in the Wolfram demo) seems to have done the most recent published research (2008). They report different results when the balls are initially touching vs a small gap between them. You could try contacting them about their results :
https://www.researchgate.net/publication/253981162_Newton's_cradle_undone_Experiments_and_collision_models_for_the_normal_collision_of_three_solid_spheres
Best Answer
The coils of the spring are touching one another and the spring is initially under self-compression so it takes a finite force to move all the coils away from one another and for the spring to behave as you expected.
That initial part of the source vs stretch curve is real (you had obtained data whilst undertaking an experiment) and hence should not be ignored.