Q1. A transitive verb needs a direct object - that is the necessary condition. Usually sentences with a direct object can be made into passive-voice constructions.
Of the two example sentences you gave, Sentence #1 is actually OK as a passive voice sentence. It just sounds slightly awkward in the passive voice. In your Sentence #2, you have incorrectly conjugated the verb in the passive sentence. The best choice would be the past participle in a form that is now archaic:
She was becomed by the dress. (She was made comely by the dress)
Became is the modern form, but rarely would you ever come across became as a past-participle for become because of the potential for misunderstanding - She was became by that dress. Similarly, you would never use became as the simple past tense for become (i.e. to flatter) because of the potential for misunderstanding: That dress became her.
It's not incorrect, it just invites confusion.
Therefore, I think the problem is clarity rather than grammatic incongruity.
Q2. As above, I think there may be more examples where the passive sentence is weird, or where the original verb has multiple meanings which make its use in the passive voice confusing. Become is especially confusing, but the confusion drops away if you look at similar verbs (beguile, bewitch, bemuse).
Q3. As above, I think they all can be made into passive sentences, its just that some past participles are confusing. The exception would be the imperative transitive, as mentioned in the other thread.