Translation:Faster! She is waiting for us.
You can have either or both. Apparently you can also add 呢 to the end to indicate continuing action.
zai is an auxiliary verb while zhe indicates the continuous aspect. The use of zhe can sometimes be idiomatic and sometimes be classified as colloquial (source: allsetlearning, Chinese grammar wiki). According to this wiki some verbs use zhe more often than others, deng is on this list. So in Mandarin you would often say dengzhe. Ta dengzhe wo. Wo dengzhe ta. Ni dengzhe! (here they all mean to wait and keep waiting) Adding zai here means that she is waiting right now, the zhe to indicate that the waiting is ongoing. The sentences you mentioned are also correct, however the duo sentence, according to my Chinese friend, sounds better.
"Waiting on" is certainly what a waiter does, but in reality "waiting on someone" is used all the time in the US. "Waiting on the bus" for "Waiting for the bus" and "Get on the airplane" for "Get in the airplane". "Take her to the airport" instead of "Bring her to the airport". "Y'all" and "Yous guys" for "You all" or "You". Even this course at least used to use "You Guys". But both are used extensively.
These are all regionalisms which should have no place in a language course. I have lived in California and Massachusetts but never heard anyone say "I am waiting on the bus" unless, of course" they are already on the bus and are waiting for others to arrive and join them on the bus. "Bring you to the airport" maybe correct in German but not in English. In English, bring and take are dependent on the direction from the speaker and the listener. I only heard foreigners say "I'll bring you to the airport" or "Could you bring me to the airport?"
Actually in some areas "bring" and "take" are used differently to how you know it. Irish people especially seem to use "bring" for most uses where others would use "take". But "take" is more standard and what the course should use by default even if it also accepts "bring" in some questions.
I'm a New York City native, "waiting on the bus" usually means the bus is late. It has the same meaning for people or anything else. This usage is very common in NYC, but it's also used in other places.
"I'm supposed to meet Jane at 2pm so I'll wait for her at the coffee shop."
"It's now 3:10pm and I've been waiting on her for over an hour."
When a person continues to wait for someone or something that is late, they take on a subservient role. "Wait on" is also in the dictionary for "wait upon" or "to wait for."