You need to have Cookie Clicker as your active tab to let Raptr track your gametime and the browser doesn't need to have focus. If you're browsing with multiple tabs, everytime you switch out from Cookie Clicker, Raptr will notify you something like 'Cookie Clicker - 9 seconds played'.
This behavior has been tested with Chrome and Firefox. With Explorer I'm under the impression that gametime isn't tracked at all.
Dungeons are a work in progress feature and are normally only available in the beta version of Cookie Clicker. However, it's still possible to earn these achievements in the regular game. Here are a few ways to get it, ranking from least dishonest to most dishonest:
Earn the achievements in the beta and import them
Because newer version of cookie clicker can import saves coming from older versions, if you play the beta and earn the achievements there, you can import that save into the regular game and you will have the achievements.
To earn the achievements, you'll need to unlock the factory dungeons first. This requires 50 factories. Once you attain that number, the dungeons will be unlocked automatically. Click on "Enter dungeons" and you'll be granted access to the dungeons. Click the arrow keys to control your hero, whose stats depend on your CPS.
The first two should be easy, as the Sentient Furnace and the Ascended Baking Pod are bosses which appear on every level. The other two will be trickier since they are rare and will appear randomly based on chance. Remember to explore all the rooms as there's a better chance of them appearing the deeper you go into a level.
I suggest you enable auto-explore (enter Game.Objects.Factory.dungeon.auto = 1 on the console if it's not set to auto-explore already) and leave the game running.
Unlock the dungeons in the live version
If you have made significant progress in your game, you might not want to lose all your progress. Luckily, you can unlock the dungeons in the regular version of cookie clicker. There are a few ways to do this. The easiest way is to open the Console (F12 on Chrome or Ctrl+Shift+K on Firefox; varies from browser to browser) and enter the following command:
Game.Objects.Factory.unlockSpecial()
Once the dungeons are unlocked, you can earn them like you would in the beta.
Cheat the achievements in
Simply open the console and give yourself the achievements by entering the following commands:
Game.Win('Getting even with the oven');
Game.Win('Now this is pod-smashing');
Game.Win('Chirped out');
Game.Win('Follow the white rabbit');
This won't give you the "Cheated cookies taste awful" achievement.
Best Answer
"Just Plain Lucky" is awarded at random: each second that the game is running, you've got a 1 in 500,000 chance of getting the achievement. It could happen as soon as you start playing, or it could take weeks. (Because the random number generator isn't truly random, it's guaranteed to be awarded eventually.)