You have to think of it as complexity vs benefit. Of course it would be great if every request was processed without ever failing, however, the complexity of doing this doesn't increase linearly, but instead increases much more sharply.
So you really have to ask yourself, are you running an application that can't suffer even a single second of downtime? If you are a bank handling transactions, probably a good idea to never even have a second. If you are running a public website and you have second of downtime, it probably won't affect your business much.
Remember that to create a setup that is truly resiliant you are adding numerous complex components, the more components you add the more likely that any one of them has an issue. Additionally you are now routing requests over several layers so trouble shooting becomes more complicated.
Now all of that aside you asked the question of the difference between floating IPs, DNS, and load balancing.
If you add multiple DNS entries for a domain or subdomain then they the DNS server will randomize the responses and send traffic to the listed IP addresses. This doesn't give you failover protection, but it does provide an easy way to build horizontal scaling, and also if you do experience an issue instead of all of your traffic being lost, you will only lose a percentage. Say 33% if one of three servers fails.
Floating IPs are basically a network configuration where an IP can switch between two servers almost instantaneously. This provides failover and especially if you just have two servers running, one in active, one in passive mode. The problem is that you need to run a service on both servers that are constantly monitoring each other so that they can pass the IP back and forth. It adds fail over protection, but it also creates a potential point of failure.
Load balancing is really doing two things. One is that it can be used to create failover protection, but it's also necessary if a single server can't handle all of your requests. This way you can load balance between multiple servers.
However, again you have another point of failure, which is why people setup load balancers in pairs so that if one fails the other one takes over.
The setup with most failover protection would be a floating IP that resides between two load balancers in an active / passive configuration which are then sending traffic back to your actual webservers.
So you went from one web server to a total of 5 servers pretty quickly, and each one adds a point of failure and you are relying on software to mitigate that.
There are also no services which will be fool proof always and remember the more failover protection you add the more complex the management of your configuration becomes.
So with that in mind you may want to use a Load Balancer from DigitalOcean as that provides you with a pair of load balancers with a floating IP between them and then you can tell it which backend servers to route requests to.