How Do I Do This Without Pushing The Big Red Button?

Like demon said, as soon as you connect to anything on the iternet, you're basically givin out your ip. So "give out your ip" argument is bull. It's a common misconception coming from paranoia that has zero grounds.

That being said as far as the server goes, don't open any ports other than minecraft's, default being 25565 iirc. As long as you don't set any remote accessing methods (eg ssh) then it's technically impossible to get "into" your computer, though I won't say 100% safe simply becUse there are ALWAYS exploits for this or that. However, alsorecognize that doing so would take a shitton of effort and then ask yourself, "Who would waste their time to hack into my server?"

As far as YC's security goes, the ssh port is a non-default port, requires a ppk key which also has a 40 char pass phrase to use, as well as a standard lengthed 16 key passcode to log into the user on the box itself. So yeah. Have fun getting through that. xD

Oh i think this mite also be a good time to mention. Hosting a server takes money. I don't know how much and it may very depending on how you do it and weather you rent or own the server. So you'll need either paypal or a credit card. the only way to run a free server that i know of at least would be going on single player and opening the world too LAN. but even then you can only play with people in thee same house as you.

You can host a free server on a home computer, but this means that people have to enter your IP to join and it might lag too, depending on the hardware.

If we didn't have a mumble, a hostname, a website, or any of the other stuff other than the ACTUAL minecraft server itself, I coudl run things for free.

It's just, yeah. Webhosting, domain registration, VPS renting, and all the other stuff does cost money and we do use them xD About $100 a year or so.