Hi,
Please understand that HikaShop 3 is the biggest update of HikaShop ever. The whole frontend of HikaShop has been modified and improved.
It's like going from Joomla 2.5 to Joomla 3. That would normally require a whole new Joomla template or at least some work to adapt it.
Furthermore, we did all we could to be as backward compatible as possible to reduce the burden to you.
The problem is that the more you had customized HikaShop, the more you'll have issues with the update. Because the issues are not in the update, but in the customization that needs to be updated to the new display of HikaShop 3.
Now to answer your points:
1. The change you see is caused by the custom CSS you have. In the HikaShop configuration, under the "Display" tab, change the frontend CSS file to "default" and that should help. It's also possible that the CSS issue comes from the template CSS so it might require some further modification. In that case, please provide a link to the account page and a test user account so that we can look at it.
2. 3. and 4. We've completely changed the display settings of the custom fields in order to be more fine-grained and clearer. Edit them via the menu Display>Custom fields, and check the display settings so that they are like you want and save them. That should be enough.
5. The wishlist interface too has been completely redesigned. The modifications on the cart system were too important to keep the compatibility with the old views so we had to change the view being used there. So your customizations are not taken into account as you could see. You'll have to redo your customization to the new view file if you don't want the default display.
6. Regarding the filtering, it looks like a double problem. The URL /producten-diensten/uitvaart-pakketten/product/listing on your website is not valid. Apparently, the link was working with the default settings of the HikaShop configuration before even though it was not a standard behavior. The issue is actually that the filter module shouldn't change the URL of the page when applying the filters, or it should redirect to a products listing page which displays all the products and in your case you have neither behavior.
So I'm feeling that the issue might come from a view override of the file "filter" of the view "product" which might have some old code to handle the submit of the filter form. Check in the Display>Views menu if that view file has an override and if it does, try removing it (after saving your modifications of the view file). That should hopefully be enough to fix the problem. If not please provide a screenshot of the settings of your filter module.