My Forerunner 405 is being replaced with Forerunner 10. When I connect my new watch will all of my old data still be there and it continues as usual with my new watch or will I have to start a new MyConnect?
If you are talking about your Garmin Connect account, then all of your old data will be there.
Data (i.e. activity history) specifically on your watch will not be transferred to the new watch. Also, any user setup like height, weight, age, will likely need to be set in the new watch.