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Akeeba Backup will back up PROCEDUREs, FUNCTIONs and TRIGGERs by default. I added this code ten years ago :)

But it is an option which can be turned off. Before going on a wild goose chase go to Akeeba Backup, select your backup profile, click on Configuration and find the Database Backup Engine row. Click on the Configure button next to it. In the pane which opens below make sure that “Dump PROCEDUREs, FUNCTIONs and TRIGGERs” is checked.

If this is already checked, it may be that MySQL itself doesn't allow you to either dump OR restore the triggers. Each trigger has a definer which is a MySQL user. If the definer is different than the MySQL user your site runs under MySQL will refuse to let it dump its definition. Also note that the MySQL user your site runs under must have the TRIGGER privilege granted to it. Otherwise MySQL will refuse to create, drop, execute, or display triggers for that table per the MySQL documentation page I linked you to.

Likewise, creating triggers also requires the TRIGGER privilege. Moreover, the privilege must remain granted for the trigger to function when making table operations per MySQL's documentation.

As for auto increments, they are definitely backed up and restored. They are part of the table's DDL. MySQL includes it when we are showing the create command of the table and executing it is a non-privileged operation. This has been present in Akeeba Backup since it was first released as JoomlaPack all the way back in 2006. I believe that on the tables where you have a problem you don't have an auto increment column but are instead relying on a trigger.