In stars (HB stars) the usual path is
then, before the Be can decay back into two He atoms,
In the early Universe, the density and temperature are dropping rapidly and there is only an instant when everything is right for the triple-alpha reaction. This is too short to produce any Carbon or heavier elements.
Yes! Although we have not identified any stars made of ONLY H and He, the oldest stars in the Galaxy are all very deficient in the elements heavier than He with the current record holder down by a factor of around 30,000 (!).
SN are like a production and delivery system for the elements.
This works up to around Bismuth at atomic #83 and is though to occur in SNI and also in AGB stars during the thermal pulse stage.
There is some direct evidence for the S-process occuring in some AGB stars. Technetium with atomic #43 is an S-process element that has a radioactive half-life of years. It has been detected in AGB stars that are MUCH older than that! The only thing that could be going on is the production of Tc in the star and then mixing of this to the surface via convection.
The R-process occurs only (we think) in SN and mostly in SNII. The evidence for R-process occuring is less direct. First, we see elements like Gold which are thought to only be produced via the R-process. It is also true that if we look at the oldest stars in the Galaxy, which were formed after only one or two SNII (these come from massive stars that have very short lives) had enriched the interstellar medium, the abundance of Iron is very low, but the abundaces of R-process elements are only moderately low.
If we look at the Crab nebula which is the expanding remnant of the 1054 A.D. explosion we see processed material from deep in the star that went SN, but mostly this is the result of the equilibrium fusion in the ``onion skin''.