1. Nope, I’ve never experienced stains from heating but then again during the winter, I heat my dolls first before clothing. I’ve never heated my dolls while they have been clothed.
2. Yes, but not from a heating blanket. I had my doll clothed in the sun. Had her dressed in the typical dark/gothic clothes and didn’t think 5 or 10 minutes in the direct sun would be a problem. She heated up rather quickly and the top did stain – that was the area the sun was really beamed on.
3. All the reported issues so far have more to do with the dyes in the clothing than the doll itself. The dye may be soluble by the oils in the TPE (mineral oil, etc), or it may simply be the excess dye common in new clothing (which leaves stains on people, too).
If you wash the clothing first, under conditions (hot water for materials not damaged by that, etc) that would help release excess dyes, it may help minimize future problems (it won’t’ stop them if the issue is not excess dyes, but rather oil-soluble dyes). Be sure to wash different colors separately or they may swap dyes.
Also, some materials don’t retain dye as well as others (synthetics often shed dyes easier than cotton, linens, etc).
Some materials (and dyes) also have a “set” process that helps them stay in the cloth better, which may not be being done by the cloth or clothing manufacturers (which are probably different places, but both much more concerned with not spending any money directly involved with getting the cloth/clothing made and sold, than whether it causes any problems later on to a consumer).