My - as a citizen and resident of Ukraine - understanding is that your questions are set a bit wrong because media coverage is incredibly messy and biased.
No, Americans are not in any special danger in Ukraine in Kiev (even here in Kharkov). For a record, an American is heading Ministry of Finance at the moment and this is an excellent reflection on role of Americans in Ukraine today. No, NATO is not going to take any action here, it is highly unlikely.
The problem overall here is that the entire country keeps sinking into full scale financial collapse, specifically related to energy, in winter time, with average daily temperatures below zero Celsius... That is, the top potential issues on the list are emergency power outages, financial and transport deadlocks, related social unrest and increasing crime levels. It does not take to have an invasion in order to get into trouble of troubles.
I realize that getting this job might be something you really long for, however considering the events here it might be a good idea to hold on until at least warmer time.