Do I need to get all puzzle pieces in Braid?

For example in the first level I got like 10 out of 12 pieces, so I didn't complete the puzzle, but I could still go to the next level.

So how do the puzzles work?

Can I get them all into one level, or do I need to go trough the next levels and the come back with magic or something to get the first ones? :)

Because on the 1st level there are 2 pieces impossible to get..

Solution 1:

You can progress to the next level whenever you reach the end of the current level (no matter how many puzzle pieces you got).

However, to progress to the final level (up the ladder in the house) you will need to complete each puzzle board as each board completed adds another piece of ladder that you will be able to climb.

On the 1st level there are 2 pieces impossible to get..

There are no puzzles in Braid that must be completed using something you collect in a later world, all pieces can be collected without ever visiting another world, you just need to figure out how, as some of them are very tricky.

Solution 2:

You cannot get all the pieces in level 2-1 (the first action stage) until:

you have the two "girder" pieces of the World 2 puzzle, which, if I remember correctly, are located in 2-3 and 2-4. Once you have them, you can use the puzzle itself as a platform in 2-1, as well as in World 1 (the house).

As I recall, every other puzzle piece in the game can be gotten without completing later levels, and you need every puzzle piece in the game to reach the game's last level.