* Time records (with a specific target time).
* Saving your own replays (hopefully automatically saving the replays of each level’s fastest completion).
* Online high scores, where you can watch the replay of the person completing the level.

Regarding the option of removing lives, I like the way games like donkey kong country does it, in that game you have lives, but when they run out you can return to your last save point, and save points are only available every 5 or 6 levels. This means that running out of lives has some consequences, but not as bad as starting all the way back at the beginning, usually only 3 or so level completions are lost.

All other choices (speeding up the level, changing the fuel drainage rate etc.) will cause some levels to be impossible on certain modes, this however provides for much flexibility.

Trapping at least x enemies during a playthrough of the game.
Completing all of the tutorial levels, provided tutorials are optional.
Setting off at least x spike traps whilst invincible.
Destroying at least x crates.
Checkpoint achievements, such as getting to level 25, level 50, level 75, finishing the game.
Possibly an achievement for dying and viewing each different death animation. Although I’m not really sure this would be a good idea, as it would reward the player for dying/failing.

On a semi-related note: have you considered trying to sell this over Steam? I know Machinarium was also developed in Flash and sold over Steam. I think it would at least be worth trying to contact Valve to see if it’s possible.

My thinking for the story text is to have an intro page before this one, with the following:

—–
[darkness]

The secrets of this unending maze are too well guarded,
its defenders too determined, too cunning.

I would need the power of flight to evade these infernal robots.

To avoid the traps and pitfalls, a man would have to pass through solid rock.

I fear I must abandon my quest.

[strip of sky, shadows of enemies passing, with sounds]

Today as I scavenged these upper caverns for remaining gems, I came upon a secret room.

I was astonished to discover obscure documents pinned underneath a skeleton.

Have I finally discovered the key to unlocking this mystery?

[cue the music, fade to title page]
—–

There’s still plenty of mystery, but a few clues about how we got here and our purpose.

Maybe just have each statue as unique collectable item. You could have a ‘trophy room’ kind of thing at the end of the game (or viewable any time, under a menu) which shows each statue you’ve obtained. They don’t have to be visually unique, just listed uniquely, so in the trophy room you’ll see a grid filled with statues, each representing an in-game statue. So for example, if you didn’t collect the statue in level 13, then the 13th item in the grid will be greyed out because you didn’t collect that particular statue.

There should be achievements too

About the title, looks good, but the font looks inappropriate and the emeralds seem a bit strange. I suggest studying some more images of emeralds, especially in relation to colour, shape and its reaction to light. Maybe also remove the text in the bottom left entirely; I think it’s best to either have no story whatsoever, or a very strong story. Having a story consisting of two sentences seems a bit silly to me. Personally, I would rather see just the map by itself, which would imply a story but leave it up to the players’ imagination.