Backup-SPSite is full fidelity, that means all associations and id are kept the same.. its a mirror copy. Import and export are not full fidelity meaning that associations are lost and no id's are kept (new guids created).
Backup/restore-SPSite is for site collection level. import/export is for site level and not full fidelity. full fidelity means you keep the workflows intact like running workflows and contnent types otherwise none full fidelity looses that link due to id's changing.
Restore a copy of the same site collection insdie the same web application
i dont get the last part? if its about what account that is going to use the workflow, the workflow would run under the account that is using it.
otherwise here is that answer:
SPD workflow impersonation step, who counts as author?
EDIT
lol hi mark, yes its my first name and my second name is not sharepoint ;). creating the workflow under that user should have no relavance. You should deploy the workflow by:
saving it first as a template, this wsp is saved in the site assets library. You need to export it (download it) to your local drive. Goto the new live site:
Site Actions -> select Site Settings -> in Web Design Galleries
section click Solutions -> click upload button -> navigate to the wsp
and add -> once loaded into the library you can then activate it:

the only thing you need to make sure is any lists libraries that it uses are already available on the live site! Also what account are you impersonating? aslong as those accounts that your impersonating exist on the live domain than it's fine, otherwsie you need to either add the user accounts that your impersonating to the live domain or update your workflow so that impersonation occures on a system account that is available on both test and live domains for active directory. i hope ive made sense ;)
let me know if you get stuck!
full tutorial here:
https://msdn.microsoft.com/en-us/library/office/jj819316%28v=office.15%29.aspx
your lucky its on 2013! as this method never existed on 2010 where you had to change its template and redeploy using SPD! the process is simple!