Well, this is a rather obvious development (in retrospect, that is). We had an $sp^3$ carbon with tetrahedral bonds, and it made diamond. We had an $sp^2$ carbon with trigonal bonds, and it made graphite. What if we just had a linear $sp$ carbon?
Such thoughts have been around for more than quite a while. Theorists swarmed around that non-existent tree. No wonder that people started trying to actually obtain that linear carbon.
Not knowing anything about you, I still can claim with a good deal of confidence that the first reports of its synthesis are quite probably somewhat older than you (or me, for that matter). Then they were proved... not quite reliable. Some groups agreed with that, some disagreed, a great deal of confusion abounded, then more reports came and things became hairy. So they stayed for the past few decades and will stay for a while, I guess.
Now to the Q&A.
- Does this compound exist?
· Define "exist", otherwise no meaningful answer would be possible.
· Define "stable", otherwise no meaningful answer would be possible.
- Is it thermodynamically stable at any P and T?
· Most researchers agree that it isn't.
- Is it electrically conductive along the strands?
· Most researchers agree that it is.
- Is it made of double bonds or of alternating single and triple bonds?
· The difference is less than you seem to think, to the point that it might not be a question worth pursuing.
- What is its crystal system?
· Nobody knows. Chances are there are polymorphs with different crystal systems.
· Nobody knows, as with any non-existent compound.
- Should it be called a polymer?
· About as much so as graphite and diamond, which means: No.