Actually, you need to play through 4 times to get al of the achievements.
Playthrough #1 unlocks 2 ally achievements, hardcore mode, power gamer (lvl 50), standard stuff, codex, and 1 given set of use-specific commands (techs, biotics, weapons).
Playthrough #2 is a new-char playthrough (not playthrough #1 character) on hardcore mode. Unlocks 2 ally achievements, insane mode, 1 given set of use-specific commands (tech/biotic/weapons), and the hardcore achievement.
Playthrough #3 uses playthrough #1 save data for insane mode (since this would be the easiest). Unlocks the last 2 ally achievements, extreme power gamer, and beats the game on insane mode.
Playthrough #4 is a new char, on any mode, that finishes the tech/biotic/weapon sets that were not finished.
Therefore, you must play through roughly 3.5 times to get all the achievements. The issue is that to get the Extreme Power Gamer achievement, you must use an old char save, and use that through the game on one mode. Because of this, it prevents you from using a new character to do more of the "use a given power X amount of times".
No, you can get all of them on three playthroughs. You need to play as a soldier, an engineer, and an adept to do so (the characters that combine classes won't work), but all the weapons/techs achievements are available within three plays.
Also, I've gotten the ally achievements for Wrex and Garrus while doing all the metal/tags/writings assignments without any issues. To be safe, you need to keep the same two characters in your party from the time you find them until their achievement unlocks, whenever that might be. Strangely, when my Turian Ally achievement unlocked, it was just before I went to Virmire. However, Wrex's achievement didn't unlock until I was nearing the very end, even though I had been using him for the exact same amount of time that I had been using Garrus.