Hello John,
If you have skype I would be more than happy to answer all of your questions or concerns, I will try to summarize it a bit here.
Since you are going to work near Taj mall, that's mean Abdoun area which is one of the most expensive area in Jordan ... now budget of 1000$ would not be realistic in that area even if it is un-furniture you need to start to think of 1000 JoD which is almost 1400 USD, so you are as most expats looking for furnished apartment, I would suggest to try maybe other areas such Shemisani or 5th circle. Paying monthly or yearly it is all based on the agreement with the owner.
Utilities costs are depending on your usage, but will not be more than 60$ (electricity, water, internet) if you are alone and if you do not use air conditions
(you do not need to use Air condition anyway).
For cars, you can easily find a car here used by other exacts, so no need to ship anything from outside, I can help you with this as I know so many expats and I might able to help you with that.
most local have good command of English including taxi drivers (at least understandable) and you can communicate easily.
Amman is super safe, it might be safer than most European cities .. no worries. Night life here is great and can be on up to 6:00 AM in the morning.
Police here are super nice, all of them are well educated, helpful and will treat you as Jordanian, all of them speak English.
There is no taboo for Jordanian women getting involved with foreigners, it is all depends on the person/ family ... if women don't want to involve, they will let you know nicely.
I just saw one of the questions, I forget to answer:
Yes, women can travel alone
they can do whatever they want in fact as men. they can party till 6:00 AM too ... as I mention, it is up to the person whether Male or Female.
No worries at all, you will feel home here.
Cheers,
Tareq