I find those forms very unnatural, and even though some seem to be somehow used, it might be a mistake…
間違え as a name has no occurrence in ALC, so I find this name suspicious.
考え as a name is quite common. I found many occurrences preceded by a な-adj, like "否定的な考えをする", and also expressions "そんな時には、ヘタに考えをするよりも" and "というような考えをする人".
For 逃げ, it's mostly compound nouns: 売り逃げ、 勝ち逃げ、 飲み逃げ, where there seems to be no related existing compound verb.
I remark that 間違え is the only of those which comes from a verb that has an intransitive version too, and which seems not to be correct as a noun. This might be an important point in the validity of the derivation of a noun.
As for the nuance, I'd say there's one as in English, when you say something like "to escape" and "to do a (daring) escape". Making a noun from the verb by using its radix allows you to give a finer description by using adjectives, without the grammatically heavy use of "こと/もの".
Also, there's a nuance between "he did an audacious tentative of escape (noun)" and "he audaciously attempted to escape (verb)", which may be renderer similarly as you suggest.
I don't understand very well your question. 連用形 becomes a noun, and it is the object complement of transitive verb する, whose subject is marked by が。
逃げをする could not be transitive. What would be its object?