Hi,
So how about turning on the "redirect post mode" setting of the HikaShop configuration ?
That will allow you to get the filter parameters in the URL of the page.
Although it won't be pretty URLs, the URLs will be usable to directly identify a particular filtered page.
And if you want to get a nice URL after that, you can either modify the router to transform the parameters in the URL into what you want, or use URL aliases that you can create via the Joomla menu manager or a third party SEF extension (which might be able to auto generate the aliases based on some criteria ?).
For your other question, the modification of the URL depending the selected variant is not recommended ; it would also generates some unwanted side effects.