Web-server – Website won’t come up on web server even though it comes up on other machines

Depends what you require your service level to be. If you need to be up 24/7, then go for a high availability solution.

Your host's current suggestion sounds a little questionable. As you've described it, if the primary server goes down, the load balancing service dies too and there's no way to offload traffic to the backup. When load balancing you usually opt for a DNS based solution that has multiple A records, each pointing to one of your servers. If one goes down, clients will try the next in the list. This means that there is no single point of failure that is within your control.

As to whether it's worth paying a lot extra for, it depends on a lot of things.

Here's my suggestions:

1. Evaluate your availability requirements. What's your budget? What's your minimum uptime requirement? Do you need all of your service to be high availability, or just some of it?
   
   
2. Research replication technologies for the types of service you are using. Most SQL database servers offer replication as a feature. It may be cheaper and more reliable to set up your own load balancing and replication across two separate hosting platforms, e.g. one on your existing provider and one on a cloud VPS provider such as Amazon EC2.
   
   
3. Look into hosting services that provide a Service-Level Agreement (SLA). A "three nines" SLA means that they are required to be up 99.9% of the time, and must pay you a penalty fee if they fail to meet it. Four nines is 99.99%, five nines is 99.999%, etc. It may end up being cheaper to go for a decent SLA rather than explicitly setting up replication, though both may be a good solution if you need very high availability.
   
   
4. Ask your provider(s) about backup facilities and the average fix response time for different hardware failures (e.g. dead disk, fried PSU, etc). Shop around to find the best figures for the right price.
   
   
5. Consider looking at providers that use data centers that operate on more than one fiber network, e.g. Cogent and Level3. This means that if one connection goes down, you're very likely to be still up via the other connection.

It's all about tailoring a solution that's right for your service. If you find that your budget isn't sufficient to keep availability at the level you need, then it may be time to rethink the feasibility of your project.

The most important part of the answer to this question was given by @joeqwerty above, I only continued the research since he could not imagine all the scenario.

Since I found that generally Dynamic DNS providers give 1 to 3 host names (host.dynamicdnsprovider.tld type) free for 30 days, then one have to buy 1 service per domaine (mydomain.tld type) each one 2x to 3x more expensive than a static IP, at this time DNS for each domain could be managed into dynamic DNS provider control panel, so I returned to my registrar which provides dynamic DNS service for each registered domain but do not support or have any Client updater, then I searched for third party client updaters on the net and finally found one shareware which can support my registrar dyn services, run in background as windows service, and point multiple domains to the dynamic IP. That way I can manage DNS records at my registrar, and use its name servers.

Now this is the second day updating, yesterday the server restarted after a Windows update, and DNS client updater restarted automatically.