1: the SKU Table is customizable, meaning you can go to Modules > E-Commerce > Classes > SKU > Fields and add custom fields, that should include it in unigrids (if it doesn't check the unigrid settings to make sure they are not selecting only specific columns)
2: You can extend most filters in Kentico, either programatically or just by editing the unigrid's xml configuration file. If you are referring to the webpart "Product Filter" there is a property "Filter Name" that you can point to a Filter Webpart (which you can build a custom one) to accomplish what you want to do. All filter webparts do is create a WHERE conditions in essense.
3: This functionality would be seperate from the filter, and you want to look into creating extensions for the UniGrid perhaps, but in most likely you'll be creating a custom web part (the textbox and button) and adding logic to find the unigrid control on the page and modify it, OR clone and modify the actual unigrid control that is displaying the data to have the textbox, button, and logic, which may be easier.
Hopefully that helps some!