Because 'divides' is transitive on the positive integers, knowing 3 divides 12 and 12 divides 60 is enough to conclude 3 divides 60.

True

True. Transitivity is exactly the licence to chain: $a \mid b$ and $b \mid c$ give $a \mid c$, so $3 \mid 12$ and $12 \mid 60$ settle $3 \mid 60$ without dividing. Chaining like this is what lets a Hasse diagram omit every implied edge and draw only the covering relations, the transitive ones are recoverable.