I was just looking into dynamic DNS servers, found Dynu.com and now I’m thinking I should be using Dynu as my nameserver for all my domains, because it seems much more configurable than my current provider.

As far as I understand, I have the following option: From my current server (which has tons of RAM and HDD, but the provider has a reputation of being unreliable), I could forward regular backups of my core pages to a smaller more reliable server, put that on a subdomain and redirect visitors to that subdomain in case my main server goes offline, right?

My main question then: How does this redirect work? At which level does it occur? How does Dynu (or anyone else for that matter) know that my server is offline? What role does the TTL play?

The way I imagine this could work internally is that Dynu changes the DNS records for the domain in case the server is offline. Is that correct? If so, is it possible to specify just an alternate IP address to be used for A records (but not MX!) when the main server is offline?

I think what you need is monitoring and failover. Take a look at DNS Failover service.