Although I do agree with Ron and Zac's answers, I would also like to give input myself to see if it may help you. First, I want to tell you what to do to assess your current wireless network, then advice on designing a new wireless network for that auditorium.
Before you make any changes, turn SNMP on on ALL of your network devices and make sure all of the management IP addresses of those network devices (not the wireless clients nor your laptop) are STATIC and not DHCP. Next, download a Multi Router Traffic Grapher. Next, make your laptop WIRED if you can because you don't want that network's wireless problems to interfere with your laptop's ability to gather information. Once your devices are in there and running, wait until the hall is crowded. When the problems begin, look at your graphs for either bandwidth saturation, client overload, etc. It is there that you will be able to assess what problem you're having. Keep in mind, you may be having more than one problem at the same time.
Now for the part that stings the pocketbook a bit that Ron and Zac already mentioned - business-grade hardware. With that, you will be able to advance your wireless network to one that can control things like how much bandwidth you will permit to each user (aka bandwidth throttling), deauth attach prevention, bonding, load balancing, etc. Prepare for a pricetag and yearly maintenance (software upgrade) costs with business-grade hardware.
Now to comment on parts of your post. You mentioned that you have a 50x50 auditorium with 200 seats. From a design perspective, here are two of the biggest factors to consider. One, your 200 seats can be assumed as 200 wireless clients. On average, depending on your hardware, an access point can handle 25 associations (25 devices). 200 divided by 25 is 8. So at a MINIMUM, you need 8 access points to meet maximum thresholds. I recommend going with at least 10 to give yourself some breathing room. Two, multiple access points in a 50x50 area can create a hostile environment if using 2.4GHz antenna's. Here's why. In America, the 2.4GHz range will support three channels. 1, 6, and 11. Meaning, each ap may have only ONE channel. So if you have only 3 ap's, no problem. If you have more, then logically you will have a channel that has multiple access points using it within range of eachother. This causes CO-CHANNEL INTERFERENCE, which results in poor wireless performance for the clients. To avoid this, get the access points that have antennas in the 5GHz range. You have many more channels so you can avoid this problem. Also SPACE OUT YOUR AP'S APPROPRIATELY. If your 200 clients are evenly spaced, your ap's should be as well.
Your 10Mb internet circuit may be questioned as well for 200 users, depending on what they're doing. Without bandwidth throttling per user, you could theoretically have a couple of users streaming and steal the bandwidth from the others. Your SNMP graph on the internet interface of your internet router will tell you clearly if 10 is enough.
To answer the question about what a wireless session is, in short it is a client connection through the access point to a destination. See https://en.wikipedia.org/wiki/Wireless_Session_Protocol. So for example, if you have a client that opens a browser and goes to Google, then Yahoo, they have two sessions taken up.