1) If this check is based on SPF, then you can modify your SPF record according to the notes in http://world.episerver.com/Documentation/Items/Installation-Instructions/EPiServer-Mail/Installation-Instructions---EPiServer-Mail-5-R2/
Otherwise it will depend on what this check is actually based on.
2) I would expect that to be the case, however it depends on the behavior of the server refusing the mails.
We tried to send test emails through EPiServerMail but the emails are blocked by our spam filter claiming that a spoof address is being used. For example the sender email address is firstname.lastname@example.org but is not via the true sender's mail server. How can this be overcome? When an email is sent to a list of recipients, do blocked emails like this get reported in bounced messages?