Is this behaviour normal?
All the products are appearing as beeing sold out, and when I checked in Index Managemenet Stock Status shows as 'processing'. This didn't happened before.
It is normal, Your indexer run when a product is stock out or you change inventory, even on an product update, it still in processing till he index properly.
If processing
stay for long time then you can refresh your indexer with SSH or try one to reindex manually from admin panel.