"Le di un beso" is correct; "le di un lavado" sounds very odd. I cannot come up with a rule on this one: I think the expression simply does not exist. Unfortunately there is also no standard rule for changing verbs into nouns, just a long list of patterns. Most of this comes from Latin, which was highly irregular itself in the formation of nouns...

We can understand both sentence, and they are really good, but at least in Colombia we usually say: "Le di una lavada al carro". The female article (una) and to get the concordance between article and verb we change "lavado" for "lavada". "I kissed her or I gave her a kiss". In this case, your sentence is completly correct and in spanish we use the same structure: "La besé". "Le di un beso (a ella)". [no es necesario usar el pronombre si en el contexto se sabe a quién se refiere].

"Di un lavado al coche" is grammatically correct. The verb "dar" requiers a complement, so it is right to use "un lavado " (which is the complement) I think the rule you refer is the one about transitive or intransitive verb

In Spanish, we say it as well . But, "Dar un lavado" I suppose it is said to emphasize the car was quite dirty and after being washed is clean. "Lavado" is the result of "lavar" Dar un beso o besar: means the same Hope it makes sense