I was on an IRC chat server and I noticed a few of the users had fake hostnames like me@fbi.gov or i@fuck.sheep and stuff like that. How did they do it? Is that just one of the features of mIRC or is it more complicated to do?

Those are probably IRCops -- people who have administrative privileges across the entire server. Being able to change their hostnames is a perk of that job.

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oddballz194@gmail.com <-- my email

Can these IRCops force you into accepting and running an application? At one point it said something like

-- Admin sent corrupt packet
-- Application keylogger loaded

It was the irc.hackerthreads.org server though I doubt the admins of that site would be sending keyloggers to people.

You'll only accept and run an app if you chose to do so. That's a function of your IRC client, not of the server.

Unless you have a client with known vulnerabilities, of course. Then it's possible to perform standard attack types, like buffer overflows.

__________________
oddballz194@gmail.com <-- my email

Would you be able to give yourself a hostname by responding to an identd probe?

Hostnames are assigned by IP address, and DNS reverse lookup. ident would only allow you to change the ident part of the hostmask -- nick!ident@host being the full format of the user ID.

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oddballz194@gmail.com <-- my email

To connect to IRC through a proxy so your real hostname doesn't show up do you have to download a separate program or can most IRC clients do that for you?

Some software has the ability to connect through a proxy. I think mIRC is one, and Chatzilla, a plug in for the Firefox web browser.

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jamato@email.itt-tech.edu <-- My Email

I heard of IRC proxies but most are banned. The chatzilla thought runs through firefox so i think a proxy on that would work

Some irc servers have vhost features, you join a channel and you do it through a bot.

*Remembers partyvan IRC days*