Poor creature, LOL!!!
Model is very nice sofar, but the rig should be redone. Have a look at Bethesda's creatures, creatures already made by other modders (Pegas Horse ranch!!!) or check out various rigging tutorials. Not all bones of the armature need to be extruded from one root bone. Most times the extremities are loose bone chains, which just get connected to the other chains by parenting.
The first bone, frome which you extrude the rest of the bone chain is prior to the following by default. Meaning, if you start your chain in the hoof of the horse and end in the head, each time you move the hoof, the complete rest of the body will move too and you don't want that!!!
Your deformations result either from not ideal topology of your mesh and/ or not exact weights. A wireframe screen of the model could be helpful concerning the topology aspect. Weighting can be done by automatic script (weighting by "boneheat" is a good choice if rigging is made correct) or manually or sometimes both. To weight a mesh you have to mark the armature, go into "Pose Mode", select a single bone and then select the mesh too. Now you can go into "Weightpainting Mode" and brush the bone weights onto your mesh. There is a color scheme from green (light weights) to red (strong weights). The more weight is painted onto an area of your mesh, the more influence the bone you have marked will have over this area.
Press "N" and the painting tools will get displayed. You can "add", "sub", "mix","blur"...)
Making such a hybrid creature is a difficult thing, even for an advanced animator, I personally would have chosen something easier for the beginning, but I wish you best luck with this one and will have an eye on it from time to time!!!
TheDaywalker :rock: