Problem 2 just looks to me like a known fact, not really a problem... and Problem 3 is only a problem if you don't accept the standard definition of "elementary", and/or if you worry too much about leftover pieces.
As soon as you can see two separate pieces inside a larger spaceship, and each of the pieces in isolation would travel at the same speed as that larger spaceship, then that larger spaceship is not an elementary spaceship.
Once you take out the recognizable pieces, there may well be sparks and other junk left over. But they might just be side effects of the interaction of the recognizable pieces, not anything that can function independently. There are way too many such fading sparks and such for each one to deserve its own name.
Minor side note -- no spaceship of any size will ever contain a Garden of Eden, because every spaceship by definition has at least one predecessor (itself).
A definition for ‘elementary’ that I think is closest to intuition is that ‘elements’ are connected (for some definition of connected) subsets of a pattern do not touch other parts of the pattern and that would survive individually in isolation.
Problems with that definition are that, after cutting of the obvious elements, you may be left with stuff. Using the term ‘elementary’ implies that that stuff either must be elementary, or be built of other elements. The former is the easiest way out, but (as an extreme example) the resulting element might be a garden of Eden with a billion cells that, in isolation, would die out. Would you really want to call such a monster an element?
Also, if a pattern of <50 cells can be constructed from a few dozen carefully placed small ‘obvious elementary’ parts, less than 100 or so time units in the past, I think many would agree the pattern’s elements are those parts.
Problem 1: AFAIK, it is an open problem to decide whether such a set of items exists for a given pattern.
Problem 2: that set may not be unique, and there may not even be a unique smallest (for whatever definition chosen) one in the set of solutions.
Problem 3: if the smallest such set is way, way larger than the pattern it constructs, and must be constructed way, way in the past, do you still consider the pattern to be constructed from those many, many elementary parts?