Let any prime be $X$
It started out with an experiment to find out if there are some patterns in primes with $X \mod 6 = 1$ Versus primes with $X \mod 6 = 5$.
I have then checked to see what are the multiples of $6$ before getting the remainders and obviously some duplicates came in due to cousin primes, such as in the case of $27$ being the multiple: $(27 * 6) + 1= 163$ and $(27 * 6)+5=167$
But here is the thing:
All these multiples shared a limited set of possible endings, they can only end in either: $1,2,3,6,7,8$ and always excluding possible endings in 0,4,5,9
Let the multiple be $Y$
Below is a short example of the first hundreds of $Y$'s generated from $(6 * Y)+1 = prime$ and $(6 * Y)+5 = prime$ while sharing the same $Y$ (I have further tested the first 10000):
Why are the endings limited to a limited amount of possible digits, and does it imply that in a probabilistic fashion, there are more possibilities for twin primes to occur than cousin primes?
If I understand your first query, you ask why $Y$ cannot end in the digits $0,4,5,9$
Looking at cases, if $Y$ ends in $0,5$, then $6Y$ ends in $0$ and $6Y+5$ ends in $5$, yielding a number which is divisible by $5$ and hence not prime, so it can't be part of a pair of cousin primes.
Similarly, if $Y$ ends in $4,9$, then $6Y$ ends in $4$ and $6Y+1$ ends in $5$, yielding a number which is divisible by $5$ and hence not prime, so it can't be part of a pair of cousin primes.
There is a similar situation regarding twin primes. Consider twin primes of the form $6Z-1,\ 6Z+1$. You will find that $Z$ cannot end in $1,4,6,9$ for similar reasons. The argument for $4,9$ is the same as for the case of $Y$.
If $Z$ ends in $1,6$, then $6Z$ ends in $6$ and $6Z-1$ ends in $5$, yielding a number which is divisible by $5$ and hence not prime, so it can't be part of a pair of twin primes.
In both instances, there are $4$ forbidden end digits, so I don't think this would form a basis to argue a difference in the density of twin vs cousin primes.