WordPress site uses MyISAM, should I switch to InnoDB necessary?

You should have switched to InnoDB at some point in the past decade, but now is as good a time as you are going to find. 🙂

mysqldump will work the same way.

MyISAM could not be backed up while the database is running either. Databases generally cannot be backed up by running using a naive method like taring up the files. The data has to be static / point-in-time consistent. So if you take a snapshot and back up a snapshot, that is fine.

With InnoDB you can also use xtrabackup, which is faster to back up and much, much faster to restore than mysqldump. You can also use mysqldump –single-transaction to back up without locking the tables (mysqldump with MyISAM results in all tables being locked during the backup process).