If you're on the desktop site, there should be a bank of characters under the text box that you can click on when you're doing a lesson. If you're on a mobile device, all you need to do is press and hold the letter until a little menu pops up.
But if you want to type smoothly on your non-mobile computer, there's this little program I use on my PC. http://www.spanishdict.com/answers/166839/how-do-i-put-accent-marks-on-my-computer It integrates seamlessly, so you just type a simple key combination and there's your accented character.
No, unless you consider the Scots-English "loch" (lake).
You can check out this interactive IPA chart for help. The symbol you're looking for is /x/ -- it's the unvoiced velar fricative. The IPA is organized along the Y axis by manner of articulation and along the X axis by place of articulation, and paired by voiceless/voiced.
You can pronounce /k/, so you have that place of articulation down. And you can pronounce /f/, so you have that manner of articulation down. Now you just have to practice the intersection.
Consider the difference between /t/ and /s/. That is exactly the difference between /k/ and /x/.
Roghchlár is a list of choices, as in a computer menu. A food menu is biachlár.
For "Irish" as an adjective, Éireannach is probably your best bet - An Biachlár Éireannach. If you were specifically creating a menu of dishes sourced from Ireland, Biachlár na hÉireann might do - it can be read as "Ireland's Menu" or "The Irish Menu", a slightly different meaning, indicating possession.