Hi, I already have done this for a project of my own, deploying the backend in Azure (beginning price is very low $0 for a shared server without domain, $10/month for one with custom domain, $5/month for a single instance of a SQL Database)
I used SendGrid for sending the emails, which comes as a bundle with Azure and its free.
I have done multiple UWP apps, even for the Xbox One, and I have done REST API backend with classic ASP.NET and now with ASP.NET Core 2.1
I use VS2017 15.9.3
I can guarantee I'll make it in 7 days, but can't charge $75 for it
Thanks!!