Yes - tons of spam! Even with many anti-spam systems in place, which means these are human spammers. I've started to ban whole countries (non English countries) because that's where most of these IP addresses come from.
In the last software and server update, I moved the site to a cheap server, and one of the reasons it was cheap was because they really provide no support at all. Well, turns out that was a bad plan. They are not helping figure out what is causing the sites to go down.
So, I contacted some people I know and managed to get recommended to another hosting company. This one costs more, but they have a full service package which includes support and troubleshooting for issues just like this.
I'm in talks with them, and will let you know when the move is planned for - although the site already goes down so much you might not even notice