We have 5 European stores and we have one warehouse, thus we would like to be able to sync the availabilities. So, once a product goes out of stock, we would like to put the quantity to 0 on one store only and the availability should sync with other stores. Is that possible?
We checked the Multi Store Sync Power app, however, it looks like we need to install it on each store and pay a monthly fee for each store...
Yes. I believe the Multi Store Sync Power app you mentioned charges the rate for every store you connect... Have you checked out Syncio: https://apps.shopify.com/syncio? I think they allow unlimited source stores.