Being part of the game hosting industry I am asked this question on a daily basis as it is one of the age old questions.
In short, Yes! You can host them both on the same server box, but No it is not advised. It is completely possible to mix and match web hosting and game servers on the same physical machine, but it is not something I would advise doing.
Game servers are something that can easily "freak" out when you begin to mix and match them with other things, even with other game servers. The rule of thumb is always to run the least amount of items on the server box you are running your game servers on, as you never know what may or may not cause issues with your game servers.
As shared web hosting account is only a few dollars a month with many web hosts, it is well worth it as to not mix web hosting and your game servers.
Now if you can't truely afford to get an extra web hosting account, than it would be better to host it on your dedicated server than not have one at all
– Web hosting in general takes quite allot of work to setup and can be very confusing, especially if it is your first time.
When you are forced to mix web hosting and game servers, always install the least amount of services in order to get your web site online. I would not suggest going all out installing all components and software that you might use down the road, add it as you NEED it.
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