Shipping via AJAX and variations problem

Hi, I need help with 2 issues.

1. Is it possible to load shipping methods in Product Page via AJAX, because I try every Cache plugin ever created and always the shipping method is cached for the first visitor who open a specific product (no matter which one), because this triger Cache the page and this is very confusing.

2. I find few days ago some problem with variations. My settings are when some variation is out of stoke, to be removed, but what I find now is that all variations are still on product page, but if you choose it, it said Currently out of stock… no errors in logs.

Thanks!