Received promotional charger from American Express to use without an outlet, should I be concerned that it will put malware on my iPhone? I'm 100% safe if I charge while my phone is powered down right?
There are articles warning against using public charging stations because there is a trust given to physically connected systems on the iPhone. So yes, the risk is not zero.
The question I'd ask myself in this specific case though is "Would a charger from a reputable org by likely to contain such an exploit system?"
I'd personally wager the odds are low and would be OK using such a charger on a personal phone, but if this were a work device with access to sensitive information I'd probably be more careful.
You could also keep your ear to the ground, so to speak, for a period of time and see if any reports come up regarding these chargers resulting in exploits, and if you haven't heard any such within a few months, the odds are even better.
You can never be 100% sure, when it comes to security. There is always a possibility of 0-day. Though, if the product is genuine, and comes from a reputated company, the risks are lowered by substantial levels. However, to answer to your second question. No, you are still not safe when your iPhone is powered down while charging. Though the odds are less, but if you have some sensitive information and you can't afford their leakage at any cost, don't use any such product. Buy genuine lightning cable and adapter.
The problem is that data can be passed through the wire, so no, unless you made the charger yourself you can't be sure it's 100% safe. In that case to reduce your risk from other adversaries, stick to using Apple chargers.
I personally would just use the charger as I don't have much trust or faith in Apple. But that's just my opinion. If YOU have trust in Apple, you should definitely stick to using their products (and only theirs)
But, I personally would just use the charger as I don't have much trust or faith in Apple anyways. - While I mind don't mind this being your opinion, it is (IMO) very ill advice in an supposedly objective answer. Recent media coverage  has shown, that Apple is very strong on securing their products and keeping them locked from third party actors. And even if they weren't trustworthy, that's still not a valid reason to expose yourself to another security risk (using the charger).  en.m.wikipedia.org/wiki/FBI-Apple_encryption_dispute– FMazApr 6, 2017 at 8:41
That's a good point -- I was hesitant to give this opinion.– DylanApr 9, 2017 at 10:12