In a nutshell, yes to all of the above.
In George Odian's book "Principles of Polymerizaiton", there was a section that dealt with cationic initiators. The jist of some work done by Joe Kennedy was that you can never actually dry solvents well enough to provide a nucleophile-free system. Even in a dry-box, there will always be sufficient water to complex with BF3 and initiate polymerization.
There is a book entitled "Hydrocarbon Resins". The most common monomer used to make hydrocarbon tackifier resins is 1,3-pentadiene (piperylene is a common name). In the "Hydrocarbon Resins" book, the authors speculate that cyclization does occur. Instead of the perhaps expected 1,4-addition reaction, the authors stated that terpene type structures form (ie 6-membered rings). A few years back I recall a Macromolecules journal article by Macedo dealing with quantifying the amount of cyclization by IR. A significant gel content is obtained with many polymerizations of 1,3-pentadiene unless very specific conditions are adhered to (indicating a multitude of side reactions including crosslinking). Some of the side reactions can be reduced by polymerizing at lower temperatures. Not familiar with any work done with beta-carotene. Much of the work on 1,3-pentadiene polymerization was also done with AlCl3 due to cost and toxicity issues.