How do turbofan internals handle the load from the thrust?
The thrust is generated at the fan and compressor blades, as these are the components which push the air back. It is then carried through their attachment disks to the shafts and from there to what are called thrust bearings. A simple thrust bearing is a fixed flange against which the shaft pushes, but of course they are usually a lot more sophisticated. Thrust can also be generated by careful shaping of the core exhaust duct, but on a subsonic turbofan it can at best be only a small amount. The force is then transferred through the engine's structural framework to its mountings and thence to the airframe itself.

Yes, everything in the engine will deform a bit under load. Luckily the load paths tend to be short, but it still has to be made stiff and strong, with as you say the mounting carrying the full thrust. Extensive use of high-strength materials is made in the engine to keep deformation down to manageable levels. Varying amounts of thermal expansion of different parts also take place. These combined movements necessitate the designing-in of mechanical tolerances, to prevent the binding of moving parts or local build-up of mechanical stresses.