How do I keep the development MySQL schema in synch with the live one?

The MySQL Workbench can do it.
There is a free replication tool named SymmetricDS, never used that though.
Then there is HandyBackup which is also free, but requires to wipe the DB and reload all data on changes. As long as the data is not too big, that's not a bad route to take as it gives you a backup and easy bootstraping of the DB for free.
There are multiple commercial solutions from affordable to very enterprisey. Notably SQLyog for $299/license.

You definitly need a separate development environment. You may know all the pitfalls of the system, a newcomer won't. One wrong step and your production crashes, potentially losing or destroying data. Related to that, you need a good set of test data and a way to wipe and repopulate the (dvelopment) DB from that. The dataset should not be too big, but complete enough it covers all (edge) cases (customers with and without an address, with or out email address, etc.) and big enough to discover Schlemils, but not so big it slows test queries down too much.