Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.) Is there then an algorithm to determine whether $G$ has an element of order 2?

If so, where is the best place to find such an implementation?

If not, what more needs to assumed? What standard methods exist to disprove the existence of 2-torsion?

Edited to add: Here is a specific example with three generators and four relations. Take $$ G=\langle x,y,z\ :\ xy^{-1}x^{-1}y^{-1}z^{-2}=1,\ x^{-1}z^{-2}xz^{-2}=1,\\ xyx^{-1}yz^{-2}=1,\ y^{2}xzx^{-1}z^{-1}=1\rangle. $$ I can prove that this group is not a unique product group, but I wish to show it is torsion-free. I can show it has no odd torsion. Any advice?