Where do you get error about wp-cron issue? I checked the site health and see that autoupdate issue https://prnt.sc/1s6ofib and I suppose that it’s caused by your rules in wp-config file, not XStore core plugin.
1) You can enable/disable scroll in Theme Options > WooCommerce > Shop > Shop page filters > Sidebar Widgets With Scrollable Content. To add color and size filters create appropriate attributes for your products and then go to Appearance > Widgets > add the Filter product by attribute widget to shop sidebar.
2) Product categories widget has the option to disable only empty categories, there is no option to choose what other category to hide there.
3) I don’t see the problem with the first tab content https://gyazo.com/e9fbaab87c674fd736a87613e6f87b96 it’s opened and visible.
4) You can hide out of stock products from grid in WooCommerce > Settings > Products > Inventory > Hide out of stock products from catalog.
5) Provide us with FTP access to your site to check this issue in more detail.
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