Local install of live site looks different
Sounds like some CSS files are not loaded on the local site. Or some additional or other CSS files are loaded on the local site.
Clear caches
First thing to start troubleshooting the local site is to clear the caches. Not only CSS/JS caches but all caches.
If the site has the Advagg caching module installed, then sometimes it is not enough to clear its cache by clicking the Administration menu link. Advagg cache might also need to be cleared by clicking the Flush AdvAgg Cache button on this path:  /admin/config/development/performance/advagg/operations
If the above does not help then on the same Advagg page there are more buttons to click, hidden behind the headings containing 'Clear' and 'Delete'.
Look at CSS with Firebug
Firebug is an addon for Firefox. Instead of this you can also use the browsers built in inspector by pressing F12(maybe another key on Mac).
See what CSS is different and if the CSS is taken from different files on live site vs local site.
Panels Page layout
Having blocks on the right vs bottom could also be affected by Panels Page layout. Check for that if the blocks are still wrong after you got all other thematic right by getting all CSS right.