I've also never seen this at universities in the US (or elsewhere in the world) but I've seen it a lot in industry. I'm guessing someone came in to 'shake things up' or they hired a consultant and are trying things differently this time. Therefore, I will answer with respect to industry which I'm guessing will actually be applicable in the end.
It is a common negotiation technique to get 'the other guy' to make the first offer with the belief this puts him in a weaker position. This seems to be what they are doing and as you stated in your question, when they make the first offer you are indeed in a stronger position.
In your last question, could you ignore their request and simply not give them a number, tread lightly on this point. Some hiring people will become angry when you don't follow the rules and will count it as one reason not to make you an offer at all (or lower your offer because you are unable to follow simple instructions).
As far as how to actually handle the negotiations, like any negotiations, it is MUCH better if you are negotiating from a stance of understanding how you can work together to benefit each other. That is, don't get locked into a 'the more I get, the less you keep' train of thought. The goal is to be creative and find a way that you can actually ADD value to the equation and then divide that new value between the both of you...leading to the win-win settlement.
The problem is if you must make some kind of an offer blind and you have no personal rapport developed with your counterpart, it is very difficult to go down the win-win path. At that point, I would see if you can find a way to change the situation and start building some kind of relationship (even a telephone call can make a huge difference with regards to finding a genuine win-win solution).
If you are stuck and you cannot have any any meaningful conversation before you give a number, then the best I can say is to make a serious statement about what you would like but make it in terms of ranges (say between $100 and $120 [replace with reasonable numbers for you] per month depending on the rest of the details). By offering a range you have retained some flexibility but allowed them what they demanded: Something.
As you might guess, they may lock in on the lower number. However, you have not committed to that lower number firmly because of 'the rest of the details' you included.
Congratulations and good luck!