For your first reference, Magnus, Karrass and Solitary (or the paper of Kapovich and Myasnikov where they re-do Stallings foldings in precise detail).

(This is probably related to the $F_2\times F_2$ example, although Martino and Minasyan do use properties of hyperbolic groups in their proof. However, their proof is pleasantly short, and the properties used are pretty basic (e.g. an element has finite index in its centraliser, etc.) so it may generalise.)

If $N$ is a finitely generated torsion-free normal subgroup of a hyperbolic group $H$ such that $H/N$ has undecidable word problem, then $N$ has undecidable conjugacy problem. Therefore, Rips' construction gives many examples of f.g. groups with decidable word problem but undecidable conjugacy problem. (However, the groups are not f.p., and their "natural"-ness is debatable!). Reference is: Theorem 1.2 of A. Martino, and A. Minasyan. "Conjugacy in normal subgroups of hyperbolic groups." Forum Mathematicum. Vol. 24. No. 5. De Gruyter, 2012 (doi).

If you are willing to assume that the word $w$ is a proper power, so there exists some word $v$ and some integer $n>1$ s.t. $w\equiv v^n$, then there are some pretty strong restrictions on the shape of possible diagrams. The starting point is B.B.Newman's Spelling Theorem, which says (roughly) that any disc diagram must contain a cyclic shift of $W^{n-1}$ on the boundary (or its inverse). Hruska and Wise went much further in their paper "Towers, ladders and the B. B. Newman Spelling Theorem", J. Aust. Math. Soc. (2001). "Towers" is the method of proof, "ladder" is the shape of the diagrams.