If you are officially legal on the French side, I don't think it would be a problem if you live "un officially" on the Dutch side. I said "un officially" because you won't be legally a resident of the Dutch Side since your paper will allow you to live legally on the French side... Sounds complicated, but it is two different country, that's why. I know many of french Legal Resident who "reside" un officially on the Dutch side and even sometime own property there. I was one of them for many years and never encountered any problem in this regards. Legal resident of one side have freedom of movement on the otherside. As long as you belong to one side, they really don't care (IMO) where you really live physically. NOw sometime, some professional contract impose for some reason the obligation to "reside" (live physically) on a particular geographic area for substancial financial aid. I though it may have been your case. If your husband employer don't really care where the house they are paying for (or at least a part of the rent) is located, then I don't think you should have any problem.
Now about the driving license... well, the island is small, but public transportation, even though it improved over the years, is not a perfect system, and the distance are not so short. You always can have the solution of using a scooter, but I wouldn't recommend it, since it is not really sfe to be on two wheels on sxm roads, and that scooter are being stolen pretty easy. If you have the opportunity, I would recommend that you learn how to drive and get a driving license. It will be also easier for you to bring (and pick) you child to school. But after it's all personal choice. I cannot see myself living here without a car.. Barbara ..do you? (I know you like to drive over the hill when that's rush traffic hour