Different Altera DE2 boards do not behave the same way

You can either use the provided core or write your own.

The C bit is probably for use with a processor core, so you almost certainly don't have to use it with C. What the core is written in doesn't matter either as it's only the function you care about.

You need to either use a soft core processor and connect using whatever bus system Altera provide (and maybe use C), or write your own module and connect to the necessary "pins" on the module to control it. In the datasheet you linked to it gives details on the registers for control and data. You would access these through the address and data ports on the module.

It may be useful to find an example RS232 to give you an idea of what is necessary. Take your time, if you are new to this as it may take quite a while to get up to speed.
Pong P Chu's "FPGA Prototyping with Verilog Examples" is a pretty good book, it starts from the beginning and works up to things like UART implementation with example code, etc. Also I'm sure Altera have loads of examples on their site (probably the DE2 board comes with some too)

Here is a decent run through of writing a UART in Verilog. It is a few pages long so you have to click on the "next" link at the bottom of each page. At the end it has a link for the full code.

Note that in order to design something you need to have a good idea of how it works, otherwise you are flying blind. If you are not familiar with the inner workings of UART then have a read up on this too (the above link goes into some detail)

Close the programmer and power cycle the board. If that doesn't work, I've also had success using the Windows Reinstall Driver function.