So, to answer the direct question of what the passwords do, they store (or restore):
Nothing more than this is captured, and nothing more than this is affected.
"Why" questions are a lot harder to answer. Well, if it's something that happened by mistake, that's not a difficult answer, but this is deliberate. I plan to write an article about this a little later, but I'll try to give a short-ish answer now.
Probably the most agonizing thing about game design is that every decision is a trade. Anything that I think will make one player feel good will make someone else out there feel bad, and I don't say this as if it's a numbers game trying to maximize the number of good feels, just there is a constant wringing of hands while thinking about which benefits I think are important enough to justify the detriments that come along with them. What I should keep, what I should give up, what I should change...
So, with that preface I'll try to explain why my passwords are this way. I wish this forum had spoiler tags. I'm going to put the following text in tiny font so that anyone who doesn't want to accidentally read it can hopefully skip it. Sumez you might want to play the game a bit more before reading this, but it's up to you:
AVAST! SPOILERS BE LOW!
I wanted to make a game where you start over ever time you power on. You don't accumulate power gradually like Metroid. You aren't granted incremental progress for every minute played like with XP in an RPG. I wanted that familiarity of starting again at the beginning. (The randomized character appearance is also meant to reinforce this, a small reward for coming back.)
I also wanted you to be able to chose your path, so that you can try different things each time you play. In this respect, there are 6 episodes of this game. Each one is its own a loop of finding a lizard, then finding its corresponding boss. These episodes are overlaid on each other so that you can spend a while getting oriented, learning the space, learning the rules, and planning out your route for each boss. All of that learning is stuff you don't need to encode in a save; once you remember it well enough, getting to where you want to be is fairly quick.
The passwords are for this episodic game. Being able to go back to a place with the lizard you had will let you make incremental, saveable progress toward each of those 6 goals. Each of those episodes has its own "mini" ending, but I also felt it was valid for the player to consider it the actual end of the game.
On top of this episodic game is the "full" game where you beat all the bosses and continue on to get a "final" ending. The primary intent is that this was for people who master the game, a new way to play once you've gotten good at the 6 episodes. Some people don't discover this for a little while, but I'd say most people clue into it fairly quickly. The passwords do not directly help with this long game, but a lot of people will expect that they should. The basic need for the password was for the player that has a hard time completing a single episode in one sitting; the "advanced" player will probably outstrip this need rather quickly, and at that point they might start to seem useless.
The way I did passwords is maybe a bit more subversive to a lot of people than I intended it to be. Generally games made now always have a save feature, or else they are intended to be played through in maybe 30 minutes tops. People have developed a lot of expectations about this. Super Mario Bros. 3 would absolutely have a save feature if it were made today; you can easily compare New Super Mario Bros. as a similar game with a similar length, but obviously has a save feature. As a result, there is a lot of potential for a traumatic dissonance when a player of my game realizes their expectations were thwarted.
I think there is some good in it, and some bad. It will for sure feel very bad for many players at least momentarily, but I hope that most can recover from it. For some that recovery itself will be enjoyable. For others this will be a wound that never heals, and that was a choice I made. As you have already mentioned, you expect you can beat the bosses again pretty quickly now that you know where to go: this was an integral part of this design. If you're willing to experiment a little with the passwords, you might even discover that you could use them multiple times to go straight to the bosses. I very much did intend for this game to work by starting over every time you power on the machine, and I tried to make the design conducive to this, even though that is not how most games are made today. That's partly why I wanted to make this game; I wanted to experiment with this concept as well as others. Restarting from scratch is a core component of the game as intended, but I also did it with the full expectation that this will rub some people wrongly.
Going back to the SMB3 vs NSMB comparison, though, look at the difference in Warp Zones. In NSMB they're kinda vestigial, missable, unimportant, not very useful. In SMB3 they're so important that everybody knows about them, we pre-emptively tell each other about this secret because it's such a rewarding feature to be able to use. You don't need warp zones in NSMB because you have saves, and can go backwards through the map. I guess this all applies even as far back as Super Mario World, where already just having a warp feature was kind of boring and unuseful, so they added multiple layers of things just to add new interest to the warp zones, i.e. unique and weird star road levels + the ultra secret SPECIAL challenges.
The passwords in my game are sort of a low-key warp zone. Something that's not a secret, but figuring out how to use them effectively might (or might not) be an interesting challenge/experiment for the player. I also have cheat codes. For example, there's a diganostic screen cheat you can use as more or less "a password for coins and bosses" if you want to. I consider this not just a debugging feature, but a critical tool for anyone who doesn't want to play the game by its base rules. If you're mad at the game for not saving your stuff, please take control. However, this stuff was deliberately left out of the manual, too. I put the cheat code information out into the world (and it's findable) but making people find it through side channels was something I thought was important. The manual and descriptions of a game are very much a part of it. (I also have the expectation that few people will even read the manual, but that's a whole other discussion.)
There's also emulators and savestates, of course. Pretty much every emulator has this feature, and even the AVS and Everdrive and PowerPak have it. Once again, this was fully in my mind while making this game. It's intended to be completely playable without such a feature, but I think it's perfectly valid to use it if you want it. Again, also very important to me that this is not an explicit in-game feature, though I did implement a weaker version of savestates, more of a suspend/resume feature for the PC version, because I did not want to punish those who wanted to play the long game in sessions by making them use an emulator instead (and most other gaming platforms have a suspend/resume now). The stock NES version will lack savestates, of course, but there are still cheats at least.
Ah, I didn't mention the coins. Well, I hope it's obvious that I designed them to provoke that question about what they do, or whether they do anything at all. I can't say whether you'll be satisfied by an answer to that question, but I think the stuff I said above might obliquely shed light on why they are the way they are.
My next major NES project has already begun, and it's going to be pretty much the opposite of Lizard, in these respects. Don't want to say too much about it, but it'll be a direct and obvious action game, no exploration, no mystery, not so experimental.
HARK! SPOILERS ABOVE!
BTW if there's a question you need answered about the game, I feel like it's much better to discuss the game on a forum with others than have an official response from the author. I don't really want to establish an "official reading" of the meaning of Lizard, but if you ask why I did something that is necessarily an answer about myself and my own interpretation of it. If you do find yourself with an important question that you eventually find an answer to, maybe consider sharing it with others on forums, or maybe submitting to GameFAQs or similar place. I definitely think this game would benefit from a well written guide, but I also feel like it would be inappropriate for me to write it myself. :S
Anyhow, thank you for sharing your experience. I hope to know how you feel about it proceeding from this realization about the passwords. I do not mind at all when people share their experiences of my game, negative or positive.
I worry a bit about trying to thoughtfully answer "why" questions like this because it seems to open the field for argument I don't need or want. If you do have an argumentative response to any of it please keep in mind that this game is done. I will gladly entertain new or interesting ideas, but I have had years to debate every minute detail of this game with myself (and often others) already. ;P Like I'm definitely not trying to justify my decision to anybody, if you don't like it or think it's bad, I don't really mind, to many people the game (or any particular part of it) is
bad, but I'm trying to answer the asked question of what I think about the thing I made.