I remember from my first number theory class a woman asking if every number that is the product of all primes plus one is prime. I know that this is false. I also know that at first it looks like it might be true. What I wonder is, do we know anything about the distribution: are most candidates indeed prime? Are there gaps where 2 or more candidates in a row are not prime?
A related question that maybe I should ask separately but I will first try here: in cases where the candidate is not prime, does there: 1. tend to be a prime nearby? 2. Can a prime always be generated by taking out one of the primes from the sequence (not the last one)?
You wrote in a comment that you’re aware that we don’t know whether infinitely many Euclid numbers are prime, and indeed your questions might be answered without knowing this. But we also don’t know whether infinitely many Euclid numbers are composite – at least we didn’t in $1996$, see The New Book of Prime Number Records by Paulo Ribenboim, and MathWorld still gives this as the current state of the question.
Together, these known unknowns imply that the answers to your questions are unknown. If we knew that most Euclid numbers are prime, we’d know that infinitely many are prime, and if we knew that it’s not the case that most are prime, we’d know that infinitely many are composite. If we knew that infinitely many pairs of successive Euclid numbers are composite, we’d know that infinitely many are composite, and if we knew that this isn’t the case, we’d know that infinitely many are prime.