In reply to BB95: Absolutely no issues in the 1st and 3rd scenario a you talk about. As regards the second scenario, if you only use it as a cow's tail in conjunction with abseils and are super careful to always be tight on from the anchors, again there are no issues.
However, using any sling as a
general purpose cow's tail or lanyard is idiotic. If you fall even a short distance you will very likely either snap the sling at the knot or end up severely bruised or with internal injuries
Potential problems are more likely if people use dyneema for cow's tails for abseiling BUT then start to keep permanently or semipermanently attached to their harnesses. The first issue is that because it is always there it is far easier to get into bad habits and start clipping in where there are risks of shock loading as mentioned above. The second is that all the wear is continually concentrated at one point on the sling where it is larksfooted.
It is worth adding that an Edelrid aramid sling will probably perform much better than dyneema when used in this manner. First, with an aramid sling it is massively easier to undo the knot after it has been loaded. Second, it is pretty much impossible to melt aramid, where as moving ropes can easily damage dyneema. Third (and this is opinion rather than fact) the aramid will probably last longer.
Finally if you are doing loads of abseiling and do want to have a cow's tail permanently attached then the Beal Dynaconnexion is specially designed (iirc by a Canadian Mountain Guide) for this purpose and is a really neat and super safe option.