When I want to freeze portions of almost anything liquid (including stocks and sauces), I use a covered ice cube tray like OXO Good Grips. Just spoon it into the tray and throw it in the freezer. Easy! Keeps very well, and makes it very easy to portion out later.
The OXO product is my favourite, but it's not too hard to find generic substitutes at any superstore. Try to find one with a soft bottom so you don't struggle for 5 minutes trying to get the cubes out and then scraping the sides for all the bits you left behind.
Like you said, the main benefit is control. I'd say the two main variables you're controlling for are amount of fat in the mixture and the tenderness and quality of the cuts used. Depending on the application, you might use a different mixture of meat. (For burgers, Alton Brown uses a 50/50 mixture of chuck and sirloin.)
Grinding your own could also be considered a play for increased food safety. If there are any bacteria on the surface of the cuts of beef that go into the grinder, they will be pretty well distributed throughout grind. The longer (and warmer) the ground beef is stored between when it is ground and cooked, the more chance that the bacteria could grow to sufficient numbers that they could do some serious harm to the consumer. This is why it's recommended that ground beef is cooked to a higher internal temperature than say a steak. In grinding, everything effectively becomes surface area so you have to cook a burger all the way through to be sure you've killed any bacteria.
If you're grinding you're own, you can make the interval between grinding and cooking arbitrarily short, so if you want to take your chances with a rare burger, this would probably be the best reason to grind your own. You're still running a risk in this case as any bacteria that were on the outside of your meat are now on the inside of your burger and won't be killed if the meat is left rare. You would just be trusting that the butcher did a good job of keeping the outside of the cuts you purchased relatively free from infection. Also, any food safety benefit assumes you're doing a good job of cleaning your equipment. Meat grinders can be a real PITA to clean well.
As far as the flavor difference is concerned, I would assume that to be minimal, again if you control for any difference in quality and cuts of beef that might be used. If your butcher grinds the beef and stores it cold in a case or wrapped for a day or two before it goes out the door the flavor shouldn't change enough that you'd notice it after seasoning and cooking. Oxydation would have had a chance to change the color of the meat over that period, the reason why ground beef can look brown on the outside but still nice and pink when broken up. But there shouldn't be enough time for there to have a marked effect on the flavor.
If there is any perceptible flavor difference you'd probably notice it more in a burger where you're tasting the meat by itself for the most part than in something like a chili or stew. (As an aside, you don't necessarily need to grind all the meat in those anyway as they tend to be cooked long enough to soften bigger chunks of tougher cuts.)
Best Answer
Store them as you would the unground beef. If it will be still be in date after a few days, store it in the fridge, otherwise, freeze them (separating the patties with a sheet of greaseproof paper).