There are many possible answers to this but the best way to answer this is to consult the official MySQL documentation 2.13.1.1. Upgrading from MySQL 5.0 to 5.1 for "Incompatible change" and "Known issue" items. Most of the answers to questions about the details of upgrading MySQL are available from the MySQL Manual.

No. The upgrade is done in place on a block level copy of the databases on a slave machine and any failure in the process will rollback to an untouched 5.0 copy of the databases on the master machine.

Hostnames may eventually be changed to provide more uniformity in naming, however, no change will occur during this upgrade process. Thus, at this time, no code changes will be required to update your connection strings.

The IP designation of the database for external connections will remain the same and will not change as a result of this upgrade.

Beyond increases in stability and performance you should generally not be able to detect any changes in the operation of your database. However, due to the fact that MySQL 5.1 has seen improvements in foreign key handling, greater RFC compliance and error handling advancements it is possible that web applications may encounter errors that previously they did not. This is actually a good thing, in that MySQL 5.0 did not report such errors and allowed data to be entered or deleted by the application when it should have been prevented. Application code that is faulty may now report errors and/or halt where before it was allowed to continue regardless of propriety.

In order to obtain a perfect snapshot of data the database must be taken down completely for a period of 10 to 15 minutes. Directly after that hard down time the database will be in a read-only state for 30 to 45 more minutes while the in place upgrade is being performed. Full function will return directly after the upgrade is completed in 40 minutes to an hour after the upgrade maintenance window begins.

Generally speaking no, you will not have to make changes once the migration is complete. Base post migration tasks will be performed for you to ensure that tables, views, indexes, functions and procedures are repaired if necessary. You may however need to make adjustments to application code, user defined functions, and/or stored procedures if the advanced error handling of MySQL 5.1 causes non-compliant code to fault.