Hi Tyler,
I'm a Core Network Engineer for a service provider and I'm also currently going through the Cisco Certified Network Professional tests now. So I'd like to think I know a few things about switches.
It sounds like you lack a basic understanding of network topology, but if I am wrong, I sincerely apologize for that assumption.
Cat5/6 is the most common medium in any networking environment. For example, my company only uses fiber optic cable on 10 Gig Ethernet connections, everything else is Cat6, although we were using Cat5 until a few years ago and there is a lot of Cat5 still in production.
Cat6 is a standard that was created for gigbit ethernet. The only difference between Cat6 and Cat5 is the number of twists per a specific distance in the twisted pairs. If your ports are 100 megabit ethernet you should be fine using Cat5.
Now as for not understanding how your 100 clients can work on the internet at one time through just one Cat5 cable, that's kind of the whole point of networking. Aggregating connections and simplifying architecture. Your clients send packets which are switched and routed to the destination. That's why you don't need to have one cable going to every possible destination, and are instead concentrated at switches and hubs.
If your connection to the internet is less then 100mb (which I'm absolutely sure it is, even our fiber customers don't contract pass 50mb/s), then you should be fine. This also assumes that your intra-network traffic (client to client) won't need to exceed 100mb/s.
I'm more concerned about the size of your broadcast domains. In what you describe you are running two data centers on switches, and then probably connecting to a router for your outbound traffic. This would mean that all of your clients are on the same virtual "Wire" meaning that all packets they transmit are not separated, meaning that a special type of packet, broadcast packets, are transmitted to all end points when one of the clients wants to match a MAC address to a IP address. This is going to create a large amount of overhead and congestion.
Any data center environment should really be running on VLANs (VLANs create virtual networks on top of your physical topology) both for decreasing the size of broadcast domains as well as for security reasons. You don't want a client PC to be able to intercept packets destined for a financial accounting or records server, which will be really easy to do if they are on a "dumb" switched network without VLANs.
Finally, while the one cat5 cable will be fine depending on what throughput you require, I would recommend running a second one for redundancy, but ONLY if your switches support spanning tree protocol, which prevents a packet from looping across a redundant connection. If you connect a redundant cable between to switches, they will replicate the packet over and over again, and will crash the entire network in seconds.
Also, unless the ports on the switches are crossover-X auto sensing, the Cat5 cable will need to be a crossover cable. Not a straight through as most pre-packaged Cat5 cables are. This is because your between to like (layer 2) devices. The only difference being the pin order on each end of the cable.
In closing, for this type of project, I really recommend you contract the work to a experienced network engineer.