"I saw him shot dead by the soldier." This is the correct sentence. A soldier shot him dead, and you saw it. If you use "shoot", then the sentence must be rewritten: "I saw him shoot the soldier dead." So basically, you're right and the question is wrong.

The word "grammar" means agreement, the tenses and numbers must agree. If you "saw" him then you're talking about the past, "shoot" must agree with "see", you saw him shot. B is correct. Why did you say that A is the solution? A is the wrong answer.

I would definitely say "shot" is correct, not "shoot". "He was shot dead." not "He was shoot dead." I can't see how A could possibly be correct. It would only be used as "I saw him shoot the soldier dead." Two very different meanings and requires a different word order.

Yeah... the solution from my one-dollar-app is often messed up... :(

Interesting question. At first glance, I thought you must have forgotten the rule "use infinitive after 'see' , 'let', 'make' etc." However when I looked into more detail, I found this is passive voice. I also want to know the explanation.