Ad to cart buuton doubles the quantity

Try these two ways.

1) Check hooks of woocommerce which you have added in your theme.
2) Download latest version of woocommerce and replace full plugin again. May be if all files not updated properly.

If I reinstall the plugin will all my products, tags and images will be lost?

I should delete any cached pages and deactivate the cache plugin until you have finished debugging.

Products and tags are in the database and won’t be affected by a reinstall. Images are in wp-content/uploads and also won’t be lost. However, WooCommerce settings would be lost if you deactivate and delete WooCommerce. Instead you could use ftp or your hosts file manage software to overwrite the woocommerce plugin directory with a fresh download.

If your theme has WooCommerce templates, (find out at WooCommerce > System Status) then make sure your theme is up-to-date.

The debug method is to deactivate all other plugins and switch to the default theme. If your site works now, reactivate software elements on-by-one and test to identify which is causing the problem.

Please post the url to your site.

Reinstalled completely woocommerce, still the same problem.
Also, On the shop page may admin black bar dissapears.
The only browser that work is internet explorer.
website is here:
https://paginadepsihologie.ro

It looks like your theme is version 2.1 but version 2.3 is available. Consider updating, but take a backup first.

Have you checked the hooks of woocommerce which you have added in your theme?