I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a group theorist myself I am more interested in the fundamental group than the manifold and a full answer to this question would include a presentation of the group.

Since Thurston's geometrization program has been realised, such a manifold must be hyperbolic and finitely covered by a surface bundle over a circle. The Seifert--Weber dodecahedral space might be such an example but I do not know whether it is or not. Such an example would lay bare the need to pass to a finite cover when establishing the surface bundle over a circle feature of closed hyperbolic 3-manifolds and its fundamental group may have other interesting pathologies.

embeddedtorus and no essential "immersed" torus. Examples that are atoroidal in the first sense are easier to write down (although not necessarily hyperbolic): you can use Brieskorn spheres, Seifert-fibred spaces fibring over hyperbolic triangle orbifolds. $\endgroup$