Mass-effect – Does accomplishments progress carry across multiple playthroughs

Yes. Most achievements and challenges are cumulative across all playthroughs and modes. The achievements relating to collectibles (Gotta find 'em all and Nearly there) seem to require using a single character, but you can repeat the campaign with a character in single-player mode, which keeps your skills and inventory intact, including all collectibles you've found, so you can use that game-plus to get the rest of the items you missed the first time through.

Also, the challenge relating to collectibles (My collectibles, let me show you them!) does count total collectibles across all characters, and they do not have to be unique collectibles. (Example: If you get Banoi Herald excerpt #1 with all four characters, that will count 4 toward the total of 140; you do not have to find 140 unique collectibles across all four characters.)

Note that this gives you an easy way to level up a new character: get to 90% or more on a Level 4 challenge with a high-level character, get to a checkpoint to save your progress, then load a low-level character and complete the challenge. Because the experience reward does not scale, a reward that might not change a level-50 character much can boost a level-5 character several levels.

You don't have to collect them all in one single playthrough. You can select the chapter with the missing note from the menu and get it this way if you happen to have missed it in your first playthrough.

In fact there are no missable achievements/trophies in this game at all.