Folds in Cross validation
I recommend trying both (more than once), and exploring any differences. In my experience, using the same set of folds for all models or using a new set of folds for each model doesn't make any material difference. Post if you find different!

Regarding "I'm performing first variable selection, then hyperparameters tuning through cv", maybe watch https://www.youtube.com/watch?reload=9&amp;v=S06JpVoNaA0 to be sure you are not introducing any bias.