I am not sure it is normal to use a Post Grid element for WooCommerce (you did not mention it was WooCommerce) in Elementor. I think there is already a template for WooCommerce in Elementor or your theme, so you would set the form to WooCommerce display results method, and show our form on the page using our form widget, with no special code or javascript needed.
I think that the Ajax Container needs to be .products for WooCommerce, the pagination selector I would have to check.
There are two Search operators you can use within a taxonomy (such as Product Category):
AND
OR
The default is AND and this would have the effect of making less results as you select more terms.
Conversely, OR will increase the number of results as you add more terms.
It is possible you have the wrong one selected in the field setting.
As to layout/styling. Some custom CSS can do as you want. I have had other users set the field as checkbox, but hide the checkboxes and style the terms more as tags (with border etc), but that is easy to do, if we can get this working OK.
Many other users have WooCommerce and Elementor working OK, so it should work.
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