Welcome.
I'm an American °µÍø½ûÇø. I used to live in Corpus Christi, and Texarcana / Shreveport. So I kind of know where you are coming from. I've lived in China for fifteen years.
You have posited a large number of questions. Rather than try to answer them individually, I will answer like this...
16,000 RMB is a standard wage for a manager or supervisor in China. Most Chinese get by with much less than that. It is also around the range for a foreign teacher. However, you are getting five weeks paid vacation. That equates to three weeks paid holidays and two weeks paid vacation.

If you live like a Chinese person, you should be fine.
That being stated, you will not be able to afford a car, or send your children to private foreign schools. They will need to attend local schools. Don't worry, they will be fine. You will probably end up cooking most of your meals at home and only eating out on the weekends.

Your employer will be able to help you out immensely . They can negotiate housing and other issues.
I would strongly urge you NOT to move anything other than two suitcases for each of you guys. Most American °µÍø½ûÇøs leave after three months. I am not at all kidding. I would say that the half-life of an American in China is a few months.

You will find that you will eat better here, and for far cheaper. You will find that you will not need to take stress medicine as the stress level is really low here. You will find that you will be happier and healthier. However, you will need to make some changes.
These will come over time.
You can hire a babysitter / housemaid really cheap. They can help you with your one year old. Daycare is an American phenomenon. China uses maids and housekeepers called aiyi's. Depending on the arrangement you work out, it could be from a few hours a day to a live in condition. they will really help you. Get a coworker to negotiate one for you.

You will not need to get a vehicle. Use public transportation. I have my Chinese driver's license, but I just hate driving here. It is far easier for me to take public transportation, subway, ride a bike, or hire a DD driver car.
Your school can help you get an apartment. Use local resources. Do not use °µÍø½ûÇøs to get you a place. I find the prices are about triple. Again, use the resources at the school to help you.

Like I said, I've been here for fifteen years, and my dog speaks better Chinese than I do. You need to keep at it. Eventually you will be able to get around and about.

All this being said, you should be fine. You won't be living like a rich person, but more middle-class. If you think that you can afford 35 RMB starbucks coffee everyday, don't bother coming. Instead, go and get a 5 RMB baozi and doujung. Live like a Chinese person and you will be fine. Live like a LaoWai (foreigner) and you might be hurting.
best of luck.