Does the follow-up emails get sent as a reply to the original one or all they all sent as new emails?

If the subject of follow up email sequences starts with “Re:”, “RE:’, “FW”, “FWD” (looks like a reply or forward) then follow-up emails are always sent as a reply (i.e., the previous email for the sequence will be quoted).

Can I schedule a campaign to be sent out later at a certain time?

Yes – you can.

Am I able to send bulk emails? Or are these sent individually?

Gmail Auto Follow Up supports both auto follow-up on individual emails and auto follow-up campaigns (bulk emails):
How to create a campaign with automatic follow-ups
How to set up auto follow up on individual emails

How does Gmail Auto Follow Up handle out-of-office email message reply?

cloudHQ has an intelligent algorithm to recognize “out-of-office response” email message. So it will not count as valid replies.

How does Gmail Auto Follow Up handle bounced email messages?

cloudHQ has an intelligent algorithm to recognize “bounced” email messages. The email addresses which are bounced will be automatically added to

list.