Yeah, Oxyclean can be rough on metallics, so if you need to de-crud the tinsel too, that might not be the best thing to use. The way you describe it, I wonder if it could be the remains of a disintegrated rubber band, in which case I don't think Oxyclean would be all that helpful in budging it anyway.
Have you tried soaking the tail in soapy water overnight? That might loosen the gunk enough to at least get a brush through it, and then you can separate out the tinsel and use stronger treatment (abrasives, solvents, etc) on the hair if you need to.
For the leftover adhesive, I would do as SaraMari suggests first. If the glue is super stubborn, my next step would be alcohol on a q-tip. Goo Gone would work to remove the stickiness, but I don't like the oily residue it can leave behind, especially for use on something porous like vinyl. Alcohol evaporates quickly and completely, so it's a gentler option.