Is it possible to construct a constrained tetrahedral mesh of a domain using Tetgen or similar software? What I mean by constrained is that there are some nodes or edges that are not free but are predefined by the user.
Yes, I've used tetgen to do precisely this.
If you're filling a surface mesh, you'll have to generate facet triangulations first. Then you can decide whether or not you want a constrained or conforming tetrahedral mesh.
The result of either choice is described in better detail within the manual.