"What are the things that we can't take through the security check?"
Though 安全检查 is much less common, it is still good to know and allow the full name. Not every full name for such abbreviations is that uncommon. For example, 质量检查 / 质检 (quality tests), 体格检查 / 体检 (health check), 自我检查 / 自检 (self-inspection/validation), all sound very fine. It's up to you to choose the right one for speaking or writing.
I said 我们什么不可以带进安全检查, and got it marked wrong. Is it possible? Even I agree it sounds awkward, but it's one of those exercises where I was given the individual 字 and I had to put them together, and there was no 是，的，or 东西 available, which would have made it easier. (我们不可以带进安全检查的东西是什么? is what I would have liked to say).
It's grammatically correct but somewhat unnatural. The pronoun and the main verb are too far part.
I don't think so. Postposition of clauses like 不可以带什么 is very common. I dissect his sentence this way: 我们 + 进安全检查(的时候)不可以带什么