Any time you earn an achievement an "Achievement Unlocked" bubble will come up with a sound effect at the bottom of the screen. The gamerpoints will not be visible on your gamertag online, however, until you log on to Xbox Live.
WHAT?!?! yes there is an icon that pops up when you get an achievement. as for being online or whatever you said, you dont have to be online to get achievements but some achievements are attainable only online.
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