I am interested in a free and simple MySQL tool which can offer online collaboration functionality in creating database general schema. Do you know any?
You should check http://schemabank.com/ it seems to do all you want
You could use subversion(SVN) as a way to work cooperatively on the structure of the database. It is also very useful when trying to figure out who changed what and when. It also gives you the ability to rollback changes to your code if a modification didn't work or caused problems.
I have often used SVN as a way to manage the code for databases, stored procedures and permissions for both MySQL and SQL Servers. It also provides a way to lock a file, so that no one else can edit it while you have it open.
Subversion Server: http://subversion.tigris.org/
You can then install Tortoise which can integrate with Windows to give you point and click access to your code.
Tortoise SVN Client: http://tortoisesvn.tigris.org/
You do need to have an Apache 2.2 server installed,
Here is a tutorial I found on the web which looks fairly straightforward.
Other solutions exist like sourcesafe and CVS, just depends on what you feel comfortable with.