Migrating/Upgrading a MySQL 5.1 schema to MySQL 5.7 schema
Some things, such as microsecond in TIMESTAMP, will automatically be available because the CREATE TABLE on 5.7 will use the new format.  innodb_file_per_table has a new default, so the tables will be laid out differently.  That should not be a problem.

mysql_upgrade is for converting the mysql tables and for ALTERing to accommodate microseconds, etc, in an existing mysql instance where you are updating the software, not where you are loading the data.

Coming forward from 4.0 can cause troubles with changed syntax, the addition of CHARACTER SET, etc.  But 5.1 is not that much different for what you are doing than 5.7.

There have been lots of changes/additions to mysql, information_schema, and performance_schema.  So, be sure not to dump and reload them.

Use pt-show-grants if you need to carry GRANTs forward.  Then check the results.

Do you have specific examples of things that concern you?