Hello, Jessica!
Glad to be of help!
You'll be happy to know that keeping track of the meals you've used as a family is easy; Disney has already taken care of that for you, of course! Take a look at this
PDF of the Dining Plan, then scroll to the second page. On the left, you'll see a sample receipt from a Table-Service restaurant. Note that, just below the check number and guest count, you'll see a clear balance of Adult and Child Table-Service entitlements which remain. You'll have a similar listing on your Quick-Service receipts as well. Simply refer to the most recent receipt you've received for your balance. If necessary, Cast Members can even check the remaining balance for you at these restaurants, should you have forgotten your total, or misplaced your most recent receipt.
Regarding Quick-Service locations, you absolutely can redeem those entitlements in different locations. If your little ones prefer the kids' menu choices at one location, whereas you and your husband prefer something from somewhere else, simply order what you want at each location (preferably nearby each other, to make locating a table easier!)
Lastly, you are absolutely correct in that your 2-year old is welcome to eat from your plates without paying anything additional for having done so. On page 2 of that same PDF linked above, it states: "The Disney Dining Plan is unavailable for Guests under the age of 3,
but they may share from an adult plate at no charge, or a meal may be
purchased from the menu. The price of the meal will be added to your bill."
Hope this helps to answer some of your Dining Plan questions! Feel free to send more; you know I love to help!