Q: IK flipping when going past origin.
A: This is called "gimble lock". Basically it means that two solutions are possible and software is giving you the one you don't want. You can use Skeleton/UpVector Costraint to contol the plane that the skeleton moves in. Just place a null on the side you want it to favor, then select the first bone of the skeleton (not the base), then select the command. When you move the Null you will effect the plane that the skeleon moves in. This is helpful for controlling the "roll" of an arm. If you hold the wrist your wrist stationary and then move your elbow you will see the need for this, since moving the effector of a character's wrist won't do it.
Q: IK only bends in one direction.
A: Yes, IK solvers in 3D animation products lock at 180 degrees. Believe me, if they didn't you would find them very hard to control.
Anyway, here is a couple of options, and there are probably more, but these should work.
Use single bone skeletons and constrain each base to it's adjacent effector. This way you can move the effector in any direction you want. Of course this motion is localized, but the bones will deform your envelope.
Use a curve deformation to deform the geometry of the tail to a curve. Then use shape animations on the deform curve to make the tail move.
Note: If you haven't used curve deformation here is a hint: The geometry to deform assumes the "length" of the object to be Y. Thus if you had a cylinder positoin it around the 0,0,0 axis so that it is standing up in the front view. Then use Effect/Freeze to lock it's transformations. Use Deformation/Curve/Create and place it on the curve. When translating use local instead of drag to move it in Y along the curve. You can scale and rotate relative to the curve as needed. If you want to reposition it before you start again, use Deformatoin/Curve/Deactivate to temporarily take if off the curve. Use Create again to automatically place it back again. If this feature is giving problems, practice it for a while. It is very simple with practice. Use the documetation too. Always use Deformation/Curve to make Extrusions follow curves if the curve is going to animate.
Hope this helps,
Urp