A main feature of CCC is that it makes bootable clones. I would clone your drive with CCC. All of your drive. Then boot, holding option. Then pick the copy you just made to boot from. See if your problems went away. CCC does not do block copy of the drive you booted from so whatever is causing this might not get backed up. Oh. I almost forgot. You need to partition the drive you are backing up to as a guid partition scheme HFS+ volume before you use CCC to make a clone to it. The new drive just has to be bigger than the stuff you are trying to keep, not as big as the physical volume you are cloning (unless you try block copy).
If you think you need a copy of the Applications folder, by all means do it but you don't need CCC for that task. BTW, simply making a copy of the Applications folder does not bring over all the stuff that might be sitting in /Library/Application Support or ~/Library/Application Support. Your apps might not run properly if you try to bring them back by simply copying them. For example, Mozilla Thunderbird has a huge mail database sitting in ~/Library/Application Support and bringing back the app by merely copying it might put you in a situation where you are setting up all your email accounts from scratch or possibly you might wind up in a situation where it won't even launch. I had a situation where mail.app wouldn't launch after a restore from Time Machine and I had to go looking in Application Support to delete all its files before I could get the thing to even launch. YMMV.