bottle caps can be recycled if they are single component. If they have an added seal glued into them, they can not longer be recycled as the cost of seal removal exceeds the value of the product. They are also dirty and needs washing
Caps are most often made in an injection molding process, and yes parts are put back into the stream as are the sprues, or runners from the mold.
The parts are run back to a grinder, and are blended back in at a certain ratio. Too much re-grind and the melt index gets funky. Then you have problems with internal temperatures and shrink of the final parts.
Caps are held to within .005" tolerances, and shrink variances cause big problems at the bottlers. Distilleries run up to 500 bottles per minute, application torques are pretty tight.
Post consumer, all plastics are again ground into "flake". The materials are sorted by chemical and mechanical means to separate them. Liners or "dirt" make no difference. But the polypropylene is such a small part of the recycle stream that it is rarely saved. The poly ethylene from milk jugs, and the poly ethylene terphthalate from soda bottles is the prime material being saved.
Carpet manufacturers in the Dalton GA area are the largest recyclers in the country. They use the fibers for obvious reasons.