Can I give you some reading which might help.
First, Buff's flock integration. Full of good info.
Buff Hooligans Adding to your flock
https://www.backyardchickens.com/web/viewblog.php?id=2593-adding-to-your-flock
Then, the Egg Quality Handbook. Look for meat spots and blood spots.
Egg Quality Handbook
http://www.thepoultrysite.com/ourbooks/1/egg-quality-handbook/
As far as the egg-eating hen, I've never had one so I cannot speak from experience. Some people report success by gathering the eggs often and leaving fake eggs in the nest. Supposedly the hen pecks at the egg, it does not break, so she gets out of the habit of eating eggs. I'd guess it sometimes works and sometimes does not.
As far as testing for a potential bad egg. The older an egg gets, the more moisture it loses through evaporation through the shell. The air sac gets larger as time passes and eventually the egg will float. It does not mean the egg is bad, but it is a good indication the egg may not be trustworthy. I always suggest breaking an egg in a separate bowl or cup before adding it to anything you don't want to lose, just in case. If you do test the eggs in water to see if they float, I recommend keeping them in the refrigerator afterwards. If you wash the bloom off the eggs (the protective coating the hen puts on the eggs) they will spoil a lot faster out of the frig, but in the frig they will still last a long time. And if you do try to float them, use water about ten degrees warmer than the egg. When you wash or put an egg in water colder than the egg, the air sac will contract, which causes a slight vacuum inside the egg. This can draw water into the egg which may bring bacteria with it. It's not a huge risk, but why not be that little bit safer?
Good luck!!!