Cache Being Weird?

So our site is only seems to be happy with the cache set to "auto-rebuild" Otherwise we have weird problems like categories will only display one category name on all category pages, the cart doesn't seem to be accurate, etc.

Almost all cache problems are related to ownership or permissions on files.

Earlier then 4.3.8 had some cache problem for categories and blocks.. I am on 437 sometimes when new products added they are displayed in blocks with new filling but not displayed in category new sorting..