Minimum number of nucleotides to force duplicate substring
The answer is 66: any sequence of length greater than 66 must contain some repeated substring (as you argue in the question), and there exists a sequence of length 66 where no substring is  repeated.

The latter can be obtained from a de Bruijn sequence with $n=3$ and $k=4$.  The length of this sequence is $k^n=64$  symbols.  A de Bruijn sequence is a cyclic string such that every possible length-$n$ string occurs exactly once as a substring of it.  It is known that a de Bruijn sequence of length $k^n$ exists.  If we take the de Bruijn sequence $x_1 x_2 \cdots x_{64}$, then $x_1 x_2 \cdots x_{64} x_1 x_2$ is an ordinary string where every possible length-3 string occurs exactly once as a substring.  In particular, no length-3 substring of $x_1 x_2 \cdots x_{64} x_1 x_2$ is repeated.  There are also standard algorithms to find such a string.