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 described linked below.

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

You can look at Stratified Cross-validation here. 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. Links below have some good discussion:

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

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

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 described linked below.

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

You can look at Stratified Cross-validation here. 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. Links below have some good discussion:

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

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