Is it possible to set up an email address (using Central Admin, Powershell, .NET dev, any other form) for outgoing email from a site collection in SharePoint 2013?
For outgoing email, you can set it through PowerShell.
$site = Get-SPSite https://siteUrl $site.OutgoingEmailAddress = 'myAddr@example.com
May not be exactly what your after but an easy inbuilt way (depending on what you already have set up) is to simply make a document library at the root site and enable incoming emails into that document library.
To enable incoming emails you need a SMTP server (I set one up on the same server that has SP) to create the domain. Using Central Admin you then enable incoming emails and create a domain name (needs to match the SMTP server domain name).
Then once enabled for the site collection go to your library and through library settings enable incoming emails. You can also set it up so when an email comes in you can get a notification about it (Alert Me or through a workflow)
Technet has a pretty decent walk through on how to do it all: https://technet.microsoft.com/en-au/library/cc262947.aspx https://blogs.technet.microsoft.com/harmeetw/2012/12/28/sharepoint-2013-configure-incoming-emails-with-exchange-server-2013/ (Using Exchange Server)