How to support more than 500 users at one time on a WiFi network

Your questions answered one-by-one:

Are there routers that allow simultaneous connection of more than 200 users?

YES - I'm not making product recommendations so simply search using your favorite search engine. Common brands are Ruckus, Cisco, Aruba, Zyxel, etc.

Will changing the subnet mask give me the opportunity to have more than 255 clients in the network?

YES - see more below.

Does WiFi radio bandwidth allow so many users at the same time?

YES and most decent APs have features to manage bandwidth.

Can it be solved via multiple access points? I mean one router and 3-5 or more access points?

YES, that would be the normal way to do it. AP's are limited to a maximum number of clients due to wireless protocol limits, however routers scale much higher, so 1 decent router will be sufficient for 500 users. Number of AP's depends on the actual AP(s) you buy.

Is it possible to have a DHCP server give users IP addresses from 192.168.0.1 to for example 192.168.3.255?

YES - very simple - a subnet mask of 255.255.255.0 (means the last part of the mask can give 0 to 255 numbers, for a total of 256). Out of that the .0 and the .255 are not used because of how IP routing works, so you have 254 usable. Then one gets used by the gateway leaving you 253 available client addresses. When the subnet mask is 255.255.0.0, that means a total of (256 x 256) = 65536 IP addresses are possible and after removing the .0.0 and .255.255 and one more for gateway you have 65533 available.

Since you are in a subnet behind a firewall, the only limitation that I can think of, will be placed by the router or AP (home ones can be typically limiting to the last part at 0, but even in the home market not all devices impose the limit). Keep in mind, the subnet mask is simply an IP addressing related "limit" and in private networks, it is essentially an arbitrary limit.

I've heard that maximum amount of users that may connect via WiFi is 256

That number is close, however, they will be limited as to the connection speed they are are able to obtain. The rule of thumb is 45 devices.

For example, a WiFi router rated at 300 Mbps with 250 devices connected will be limited to an average of 0.8 Mbps.

There are a number of practical limitations:

Practical Limits of Wi-Fi Network Scaling

Connecting 250 devices to a single WiFi access point, while theoretically possible, is not feasible in practice for a few reasons:

• On home networks, all devices normally share a single internet connection. The performance of clients' access will start to degrade as more devices join the network and start using it simultaneously. Even just a handful of active devices streaming video or downloading files can quickly max out a shared internet link.
• Access points overheat and stop working when operating at extreme loads for extended periods, even if handling only local traffic and not accessing the internet.
• Having a large number of WiFi clients concentrated in close physical proximity, like a home or office building, generates significant wireless signal interference. Radio interference among WiFi clients degrades network performance (due to frequent re-broadcasting of messages that fail to reach their destination) and eventually causes connection drops.

You can get around these limitations by adding additional routers or access points:

How to Maximize Your Network's Potential

Installing a second router or access point on a home network can greatly help distribute the network load. By adding more access points to the network, effectively any number of devices can be supported. However, this will make the network progressively more difficult to manage.

Source Wireless Network Capacity…How Many Devices Can Connect to my WiFi Network? - Actiontec.com

And:

A general rule of thumb is to limit the number of simultaneous connections on your home network to 45. However, the specific number is going to vary widely depending on what each of those devices is doing. For example, downloading MP3s, ISOs or other large files requires much more bandwidth than checking email or simple Web browsing. Likewise, if the network is hosting Web, FTP, or Gaming servers, the recommended limit for the number of network connections may be much lower.

Source How Many Devices Can One Wireless Router Handle?

Theoretically, you could even use a /8 network (10.0.0.0 netmask 255.0.0.0) for your WiFi. That would give you enough ip-addresses for your users. But that will not be practical.

In practice though, you will need to use an enterprise-grade solution. Most home-routers stop at round 30 users (dual band Netgears at 64, etc..) and you will have to make a routed network with more that 16 access points to make your server available.

Enterprise-grade solutions come with a price-tag. For example, an Aruba Wireless LAN controller with a few access point is easily round 10k EURO (convert to your local currency) and requires quite a bit of knowledge to set-up. Cisco is even more expensive.

Meraki is (just a bit) less expensive and offers a "Cloud" solution that is relatively easy to use, but it may not match your requirements.

The number of access points that you need also depends on the protocol that you use:

Protocol                    Bandwidth   #clients per AP
802.11b                     600 kbps    13
802.11g                     600 kbps    43
802.11n (2 Spatial Streams) 600 kbps    273

(that is Cisco, but other vendors should be comparable)

If you are going to use a stack of home-routers, than you also have to look at interference. Your 16 routers will have a limited number of bands and they will slow down the connection with other routers. Not also, that a stack of 16 home-routers is also not very cheap.

If you're going to use an enterprise-grade solution, DHCP will be incorporated in that. But get some expert help before deciding on the architecture for these size of networks.