I did notice the smoke on the ship yes, wasn’t sure whether that was just because I had been hit or not - and on that note… 
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Welcome screen star field is a bit intense, e.g. the speed of the stars whizzing towards you, additionally, you could maybe move the source of the particles around from time to time in different directions to make it look like your ship was moving around too. You could also grab the mouse position and move it in accordance with the player moving the mouse, it would be another one of those little hidden features you’ve done so well at adding already. The particles I think are a bit too big/puffy.
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Introduction text is a nice touch and has some good humour, but it’s quite long, whilst there was the option to skip I did actually want to read it, but maybe just a slightly shorter version.
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Quit Option - not sure if the text about not being allowed to quit was just because you know that it won’t quit in the browser or not, but as that option is redundant on the Web version I’d remove it (or if cross platform, check what platform and display if relevant)
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Music - love it, fits really well with the game, but during play mode I think it’s a bit too loud, I can’t really hear the shooting/explosions (sadly only me exploding) - a little reduction in the volume on the music may enable the sound effects to be heard more easily… Or… for the gold star… Options on the main menu, let the player set game volume / music volume themselves!
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The “warping” effect to the start/next level is a really nice feature, perhaps in a different colour so it stands out more against the black and white star field?
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In game star field - I think the stars are just a bit too big (as above) and a bit fluffy in comparison to the sprites used for the enemies and player ships.
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Score - ooh, now I’ve seen this on a few of the Laser Defenders and it’s slap bang in the play space, which means if I move my ship below it my shots, and more importantly, the enemy shots are harder to see… my eyes spend more time looking at the top of the screen in your game than at the bottom, as I don’t really need to focus too much on my own ship, and I can just hold the space bar down, but what I do want to look at are where the enemies are and any special ones that fly across the top of the screen - as such, a bar across the top of the play space with the score, the wave, health/# of lives etc would work well (again, different colours than black/white)
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Player and enemy laser sprites - look to be the same height as the firing ship - this seems quite bit… they do contrast well so at least they can be seen. I couldn’t really tell at the speed of the game whether they are different colours or just white, I think white, but you could consider changing the colours too, perhaps the enemy/player lasers are different colours. Maybe specific type of enemy shoots a completely different colour - maybe there’s an odd random one that shoots different colours to them all - perhaps this one gives me extra points!
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Layers - I don’t think it was intentional, but I saw that the mothership that flies across the top of the screen sometimes is above/below the main enemy sprites (in one case it went over one and under another) - it seemed a bit unusual, it could perhaps have it own “row” as it were that it flies across in at the top… maybe after a little while the enemies move downwards.
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Player stats - you mentioned about the smoke and I’ve seen some particles coming from my ship, but I have no indication of my current health other than that, first game I wasn’t sure how long I would be able to play for, e.g. one hit and I have to start again? Then I got hit a couple of times and didn’t seem to die, but I wasn’t sure if that was just dodgy collision detection (no offence!) or because I have health etc… a visual indication would be really cool.
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Player hit - how about a little flash and a sound effect when the player is hit?
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Enemy hit - how about a little flash and a sound effect (different to the above) when an enemy that needs more than one hit is hit?
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Enemy arrival - often I see enemy lasers coming down the screen before the enemies have appeared. As your enemies fire pretty quickly it might be worth giving the player a moment for them to arrive before they start shooting? You could also prevent the player from shooting until they have arrived. Would need to play test this one a bit more to see how that affected the game play I think.
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Player fire rate - it’s great fun, but, because I can shoot so fast I can clear a wave pretty quickly. I don’t really have to interact with the keyboard that much because I can hold the space bar down. You could considering limiting the player fire rate, maybe only three shots at any time, but then you could provide a power up which, maybe for a limited amount of time, enables a more rapid fire rate? You get the best of both worlds but don’t make it too easy at the same time. If you have the top bar with the player stats and so on displayed, this could be a useful area to display the “energy” that the power-up has remaining until it will wear off (assuming a single power-up at a time mechanism)
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Power-Ups! Extra lives, score multipliers, constant beam laser, rapid fire, angled burst and so on… maybe one drops from shooting a random enemy at some point… maybe a random power-up falls from the mothership if you can hit it? Maybe that just gets you 1,000 points…
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Power-Downs! Can sometimes be handy if you get stuck with something you don’t want, but equally, if you have random power-ups/downs you can really change the game play… get to a fast menacing level and drop to a single shot rate of fire!! Gaaah! 
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Mothership - saw this start coming across the screen after the text that stated “Enemies defeated” - I wasn’t quite quick enough to blast it - but then flew through it - no collision It also disappears from the play space before it has fully moved off of the side of the screen
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I dont know why a planet becomes a smiley potato?
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Enemy movement - I love the formations and the way that they appear but after that, the patterns/randomness/AI disappears, they just move side to side… you could add something simple like after a specific amount of time, if they haven’t all been defeated they move down the play space a little, no in an ongoing fashion, but just to make it harder to dodge their shots, thus it’s in the players interest to destroy them as quickly as possible, or speed them up a bit, or both… or, make them fly off the screen in the reverse fashion that they arrived, then bring them back in, but with total minus how many the player shot plus some reinforcements
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Asteroid level - I really love this… it’s perfectly placed and gives the player a little break from the enemy space ships and instead something new to think about! I would love to see those asteroids looking like they are tumbling, not just rotating sides ways but as if they were rolling towards me, different rates perhaps. Also, when I shoot the larger ones, be kinda neat if it broke into some smaller ones, and so on until you have them at your existing smallest size asteroid. Breaking apart on random directions perhaps, and if they knock into other asteroids, maybe that sends them off in a slightly different direction also. Another cool feature you could add here would be to allow the player to have vertical movement as well as horizontal - wouldn’t that just throw them after having several levels of horizontal only - would give them the ability to dodge indestructible asteroids too! 
Whilst this level serves as a change in pace and a distraction I couldn’t help but feel it needed some thing more… for example, you have a story, as introduced by your text at the beginning, so, what part of the story is this? Maybe this is the player breaking away briefly to recharge lasers/shields/energy, grab a couple of power-ups from a star base, but in the process hit an asteroid field - I don’t know, but it would be nice to in some way explain it, a bit like the hidden/special areas on platform games where you stumble across them and get all sorts of goodies… maybe it’s just a shooting range but I get score multipliers for hitting certain coloured asteroids but lose points/multipliers if I hit the wrong ones?
…then send the player back into the fray…armed to the teeth with score, power-ups, shields, lives and hungry for more…
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End of level boss - I forget the name of the game I used to play on my Atari ST, but there was one in particular where after getting through all of the horizontal sections of the level you got to the big boss… it had various parts which had to be hit and eventually you destroyed the whole thing… all the while dodging what it was shooting at you… a big boss at the end of a level on your Intergalactic Aid game would be great… maybe you blast a space aged DHL lorry, whilst little enemy ships fly out of it and the whole thing is just chaos… but reward the player significantly when they beat it
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Restart! When I play a game that is fun, I want to jump straight back into it… I don’t want to go to the main menu, I don’t want to read the story again… I wanna play! Restart and Main Menu options would be neat…
Ok, so there you go - some food for thought… nothing is meant to be a negative criticism or anything like that, far from it, I know you said you probably won’t go back to it, but if you ever did, I hope that these may provide a few ideas to stir your own creativity even further - and then make sure you share the link so I can play it!
But for any future games that I want to spend a lot more time and effort on I will add a sound screen in the options menu.