Is w ∈ L(M ) ⟹ ww ∈ L(M) co-semi-decidable?
You can show that this language $L$ is not co-RE in the following way:

Assume $L$ was co-RE, that is you have a semi-algorithm $A$ to decide given an input machine $M$ whether there exists a word $w\in L(M)$ such that $ww\notin L(M)$.

We show that we can build a semi-algorithm to decide the complement of the halting problem (on empty input), which is impossible. 

The algorithm goes like this, on input $M$:


Build a machine $M'$ that


simulates $M(\varepsilon)$ on input $00$ and accepts if $M$ halts, 
accepts on input $0$, 
and accepts on all other inputs.

Run algorithm $A$ on $M'$.


Now $A(M')$ will answer yes iff $M'$ accepts $0$ but rejects $00$, meaning iff $M$ does not halt on empty input. Hence we have constructed an algorithm that semi-decides the complement of the Halting problem, which is a contradiction (it is not possible).