#1 is correct. I will assume that you intend to say just one thing, and you are choosing between which of these sentences to say it. The phrase "if elected" kind of implies a "be" verb in there. I'm not sure how a grammar teacher would explain it, but it's kind of like we just shortened the entire phrase "if he were to be elected" into simply "if elected". I think you could do the same with any verb past participle, though it might sound odd. For example, "If grounded, he won't be able to go out with his friends." I'm fairly confident that that is correct English. It's just a hypothetical statement. It will be understood, but it sounds strange. "If elected" is idiomatic and the verb "elected" here is special because people often use that exact phrase, so it doesn't sound strange.

"If elected" is correct. In the second sentence it should be written as "If he were to be elected" Also I would change "could" to "can". I'm not an english teacher but the connotation is slightly different and "can" refers more to the ability to do something and i think it sounds a bit more correct in this sentence.

Both are correct. In both cases, the missing words "he is" are understood: "If he is elected..." "If he is to be elected..." Grammatically, the two sentences are nearly identical. In #1, "elected" is an adjective predicate complement. In #2, "to be elected" is an infinitive verbal clause acting as an adjective predicate complement. Really, the two sentences are identical, both in grammar AND meaning.