Dropping eggs in boiling water is a great way to rupture the shells and make a mess.
Not hard boiling fresh eggs but let them sit in the refrig a few days works for me.
You can use a large enough soup ladle, put two or three eggs in at a time and lower them into the pot.
I have had that same problem of eggs placed into boiling water rupturing. Maybe the trick is to let them warm to room temperature and then place them in water just starting to boil?
We have several hens (laying like crazy right now!) and keeping some eggs to age, to boil, does seem to help with the peeling.
Agreed with others that one should quickly as possible place the cooked eggs into water as cold as one can get from the tap (and running, with the new container under the faucet) to stop the cooking process, if yoke surface appearance is important.