Learn English – verb opposite of “to exist”

The nearest I found is equivalent to "be possible to be efficient/sufficient to achieve given goal." In many contexts it will be equivalent with "be possible".

avail

(intransitive) To be of use or advantage; to answer or serve the purpose; to have strength, force, or efficacy sufficient to accomplish the object.

The plea in court must avail.
This scheme will not avail.
Medicines will not avail to halt the disease.

Say, the opportunity for the final move avails.

Perhaps you're looking for "misnomer"?

This would apply to your example of the "Republican" party, at least, where a name that used to be descriptive no longer is.

It could apply to the other examples, as well, although those are more about the act of lying than about a misleading term.