That's right -- I think the reference given in the answer above might refer to existence of a trivial auto (but not endo) morphism only... I'm not sure we already have an answer for the question whether for any cardinal there is a rigid graph (in the strong sense).

$D$ does contain the constant 0 sequence because the harmonic series diverges.

That's right, @JochenWengenroth. The identity map $\textrm{id}: X\to X$ on a space $X$ is universal if and only if $X$ has the fixed point property (every continous self-map has a fixed point), which is quite a strong requirement for topological spaces. If $u:X\to Y$ is universal then $Y$ has the fixed point property.