First Sentence
She hoped that we would not construed her decision to run for office as a thirst for power.
The first sentence is simply wrong, because in this particular case "would not " requires a verb or verb phrase in the present tense. So, you could say "would not construe" (as in the second sentence) or reorganize the sentence to make it a passive verb (e.g. "be construed"), but I'd avoid the latter without a very good reason. Instead, my preferred construction of the sentence would be the second sentence.
Second Sentence
She hoped that we would not construe her decision to run for office as a thirst for power.
This sentence roughly means:
At some point or period in the past, the woman who is the subject of the sentence wanted to compete for an elected office without being seen as a self-serving person who is hungry for power, or without the decision to run being seen as a self-serving attempt to gain (unwarranted or excessive) power for its own sake.
Unpacking the semantics of the sentence doesn't simplify it, though. The original sentence, once corrected, is one of the simplest and most idiomatic ways to express the general idea.
Differences Between Sentences
The first sentence is ungrammatical. The second sentence is correct. The third sentence may or may not be grammatically correct, but it sounds stilted and unlikely to come from a native speaker.
The second sentence is one a native speaker might use. The third sentence attempts to use a construction that implies that some action began and ended in the past. So, this sentence:
She hoped that we would not have construed her decision to run for office as a thirst for power.
tries to convey the sense that the woman wished/desired (in the past) that some group of people would interpret her decision to run for office positively, but "would not have construed" just doesn't work with this sentence even if it's technically grammatical. It would make more sense when referring to a time-limited or hypothetical action in the past, such as:
If we'd known about her decision to run for office, we would not have construed it (at that time) as a power grab.
This uses past perfect tense and a modal verb ("would have") to make the hypothetical more explicit, which is what I suspect the third sentence was aiming for.
