**Is it correct to say that 10-fold cross-validation is not appropriate for these data?**

No but you have to tweak the procedure, an example is <strike> described </strike> linked below.

**What are the alternatives? Is repeating 2-fold validations 1000 times a better option?**

You can look at `Stratified Cross-validation` [here][1]. Since the data is highly unbalanced consider other options such as: SMOTE. But be sure to sample only in training data and not in the test data, i.e., sample within each fold as described [here][2]. 
Links below have some good discussion:

a. [Dealing with imbalanced data: undersampling, oversampling and proper cross-validation][3] 

b. [8 Tactics to Combat Imbalanced Classes in Your Machine Learning Dataset][4]



  [1]: https://stats.stackexchange.com/questions/89768/cross-validation-in-unbalanced-datasets
  [2]: https://stats.stackexchange.com/questions/60180/testing-classification-on-oversampled-imbalance-data
  [3]: http://www.marcoaltini.com/blog/dealing-with-imbalanced-data-undersampling-oversampling-and-proper-cross-validation
  [4]: http://machinelearningmastery.com/tactics-to-combat-imbalanced-classes-in-your-machine-learning-dataset/