Roundcube is a webmail client that your host is using to allow to read and send email on there email servers.
From the problems you describe I would assume the problems are one of (or a mix of both) the following two things:
1. The mail server thats part of your hosting is not working all of the time.
2. The server your sending email has been blacklisted as spam.
What I'd recommend is changing web hosting providers. The one your currently using belongs to EIG (Endurance International Group)
https://en.wikipedia.org/wiki/Endurance_International_Group which is widely agreed to be the worst company in the hosting industry. They buy out brands that have a good reputation and then over fill the servers and outsource the support which leaves all the customers with terrible service that will always have problems like the ones that your having. A good resource is
http://www.webhostingtalk.com/ which would be able to help find your a better hosting provider, just make sure its not owned by EIG or you'll just have the same problems again.