I have a home server which I use to give my other computers internet access. If I host my website from this home server will it let hackers access my other computers???
Just like hosting it in a data center, your network will be just as vulnerable as you let it. If you're hosting anything mission critical, it's best to host it in a data center where there are physical safety measures, redundant power, etc..
My gut feeling is to say YES. At a company you reach their web server after going through other network devices that inspect and filter regular Internet traffic. I doubt you are doing the same thing at home. If you are not, then you can consider your web server to be in a DMZ or demilitarized zone that does very little to protect the server. What you need to do is place a few protections in front of the web server such as a web application firewall (mod_security), access control (ACL), bandwidth limiting (anti-leaching), intrusion detection (IDS) and/or intrusion prevention (IPS), anti-virus, load balancing, etc. Guess it really depends on what you want to do, and how far you are willing to go to protect yourself, your data, and your customers. In a home web server, maybe you feel like a lot of protection is not required because you don't advertise it or make the mass public aware that it exists. But you cannot rely on obscure URL and nested subdirectories because you will still be at risk from automated programs and scripts that are designed to crawl web servers for any number of reasons (indexing, delivering a virus, destroying your files, copying stuff, taking over the machine for another purpose, etc).
Provided that server isn't compromised, then no, they won't be able to see the other computers on your local network (behind your nat or firewall), they will only be able to see your server. Now, if the server itself gets compromised then yes they could further penetrate into the other computers on your local network. But as has already been said, it is far better to host your websites in a place with a static IP (most home connections IP is dynamic), higher bandwidth, redundant power, redundant internet carriers, and redundant cooling. Your site will be more reliable and load faster for the world too Hosting at home is a bad idea in general.