Below are video solutions for the puzzles in the game, all within the move limit set by the developer.
I tried to play as cleanly as possible (no undos, minimise the number of mouse clicks, etc.) so hopefully they will be easy to understand.

YouTube Playlist (doesn't cover the seven miscellaneous achievements):
https://www.youtube.com/playlist?list=PLgQ2X4jgpww2XfI87Ig7A-z-kkShuD4En

G-10 probably ended up being the most difficult puzzle because of its optimisation. The later levels aren't optimised at all, so they ended up being pretty easy.

These are levels finished in under the move limit by the community. Post in the comments below if you find more!

B-10: 26/27 (830) -- by 比那名居peach, no video (image)
B-11: 17/19 (1341)

C-4: 31/33 (1278)
C-8: 29/33 (1820) -- by Bubi, video
C-12: 51/55 (1724)
C-13: 25/40 (3200)
C-14: 60/66 (1320)

D-8: 57/63 (2210) -- video
D-10: 19/21 (2210)
D-12: 51/59 (2670) -- by 比那名居peach, video + at #13, take green, lower blue, and place green left of blue
D-13: 124/142 (2748) -- by 6ap6apblckaAa, video
D-14: 271/309 (2280) -- by Bootleg Harold via YouTube comments

E-1: 25/27 (3024)
E-2: 19/26 (3829) -- by 6ap6apblckaAa, video
E-5: 87/109 (3006) -- by 比那名居peach, part 1 2 3 4 5
E-8: 79/81 (2050)
E-10: 18/20 (3108) -- by Bubi, video
E-11: 43/61 (4536) -- video
E-13: 86/94 (3496)

F-2: 82/86 (2940) -- by 比那名居peach, part 1 2 3 (84/86 video, take the red after #40)
F-3: 102/164 (5144) -- video
F-5: 16/24 (5400) -- by 6ap6apblckaAa, video
F-6: 38/40 (2947)
F-8: 72/78 (3897)
F-9: 27/47 (6960)
F-10: 109/141 (5170) -- video

G-2: 75/115 (4904) -- by 6ap6apblckaAa, video
G-3: 88/102 (4640)
G-4: 71/73 (4110) -- by 6ap6apblckaAa, video
G-5: 39/83 (8510)
G-6: 43/56 (5210) -- video
G-8: 31/41 (5290)
G-9: 49/51 (3744)

X-1: 61/107 (6318) -- by 比那名居peach, part 1 2 3 4 5
X-2: 56/80 (3997) -- by 比那名居peach, part 1 2 3 4 5 6
X-3: 66/144 (8730) -- by 比那名居peach, part 1 2 3 4 5 6
X-4: 140/206 (5890) -- by 比那名居peach, part 1 2 3 4 5 6 7 8
X-5: 121/131 (3897) -- by 比那名居peach, part 1 2 3 4 5 6 7 (above video for steps 53-103)

Score: 230,086

reky has 22 achievements, the majority which will unlock for simply beating the game.

• (8) ... levels bested! achievements
  There are 8 achievements tied to matching (or beating) the move limit in every puzzle. These will unlock after besting 30, 40, 50, 60, 70, 80, 90 and all 95 puzzles in the game.
  Note: There's an unintended bug in that last achievement unlocks after besting only 94 levels. Though for completion's sake my videos cover all 95 levels.
  
  
• (7) Serial High Scorer achievements
  There are 7 achievements for scoring a particular number of high scores "in a row". I'm not sure what "in a row" means in this sense, I tried everything to break the streak and not unlock the achievement, but they always unlocked for besting the first x levels. I tried finishing a level over the limit, skipping a level, and then going back and still unlocked the achievement.
  These will unlock after besting the first 5, 10, 15, 20, 25, 35, and 45 levels.
  
  The remaining 7 achievements are for miscellaneous things in specific puzzles.
  
• (4) Explorer achievements
  There are 4 achievements for landing in unintended locations in particular levels. The locations can be visited in any order, but the achievements will always unlock in this order: Casual > Seasoned > Veteran > Master.
  
  Level B-9
  Fall to the floor under the row of white cubes at the beginning.
  
  Follow the solution for solving puzzle until the second blue cube is lifted. Reset the blue cube and lift it again. Take the nearby portal once more to fall to the lowest floor.
  
  Level B-10
  Fall to the floor under where the blue cube begins.
  
  Same concept as the previous achievement. After lifting the final blue cube to solve the puzzle, reset and lift it once more. Take the nearby portal to fall to the lowest floor.
  
  Level C-11
  Fall to the floor under where the blue cube begins.
  
  Enter the portal next to the starting position to go to the upper floor. Lift the blue cube to the upper level then reset and lift it once more. Take the upper portal to fall to the floor.
  
  Level D-12
  Fall to the floor under where the blue cube begins.
  
  Same concept as the previous achievement. Take the portal to the upper floor. Lift the blue cube once, then reset and lift it again. Take the nearby portal to fall to the floor.
  
  
• (3) Climber achievements
  Similar to the Explorer achievements, but these 3 achievements are for climbing. Two will unlock normally, with the third requiring some extra work. Like above, the locations can be visited in any order, but the achievements will always unlock in this order: Casual > Seasoned > Serial.
  
  Level C-2
  
  Follow the solution for solving the puzzle, but continue climbing higher with the two blue cubes instead of entering the exit portal.
  
  Level E-13
  
  This unlocks naturally when solving the puzzle. It pops after entering the portal to the small platform. See step 44/94 of my solution for the exact moment (popup isn't shown in the video though).
  
  Level E-14
  
  This unlocks naturally when solving the puzzle. It pops after entering the portal after the initial climb. See step 30/106 of my solution for the exact moment (popup isn't shown in the video though).