I’ve only played the 3DS version thus far, but most of this should apply to the other versions as well.
The games themselves are exact replicas of the NES games with no modifications. All of the NES flicker and slowdown is preserved in full NES glory. Because they are identical to the NES games, you can generally consult the Game Hints pages (links above) for those games for more details on how to play them. The main differences would be things like the MM3 “controller two” cheats not being available on systems like the 3DS (for lack of a second controller).
If your version of the game is lacking a manual (as mine was), you can use the Strategies page for some tips on how to play if you’ve never played a classic Mega Man game before. Use the button configuration menu to figure out which buttons do what.
The museum for each game contains miscellaneous artwork collected by the game developers. Some of it consists of scans of physical items like boxes and stickers. Other pieces of art include concept art (including of enemies and ideas that went ultimately unused) and production artwork.
There is also a music test as well. Almost all of the songs from the six games are here, including win and lose jingles and ending/credits music. Oddly, MM6’s ending music is missing. I’m not sure why. (It’s in the Song of the Whenever...)
It doesn’t appear that you have to unlock these extras. You can access these menus from the moment you first turn the game on.
Sadly, the interface for these is not all that great, in that I can’t find any way of paging through them (other than by jumping from one category to the next in the list), so about the only other way to navigate them is to arrow through each one individually. (The database doesn’t even offer a list; you just have to find what you want by hunting through everything one by one.) This is made more cumbersome by the fact that the game tries to load each item as you put the cursor on it, which makes the cursor somewhat sluggish; you also cannot hold down an arrow button to scroll quickly, but instead have to press the button repeatedly. (On the database, if you just want to fight a Robot Master, try pressing left from the first item; the bosses are usually near the ends of the lists.)
It seems you have infinite lives while in a challenge (or at least, I was never able to run out of lives, and believe me I was trying), but which weapons and items you have available for any given section depends on the particular save state. Your health bar does carry over from one save state to the next, but your weapons do not. If you die while in a challenge, you restart at the last save state that you’d reached. The timer keeps counting down during the delay before you start, but not while the game is busy loading the save state.
You do have to unlock most of the challenges by beating earlier challenges.
Ironically, some of the challenges are made more difficult by the pure nature of the save states. Some of the challenge save states dump you into horrible starting conditions. For example, the Mecha Dragon: any good player would have been much farther to the right by that point. So you have to scramble to get ahead before the dragon appears and blows you to kingdom come. Also, the long delay before the action actually begins makes you uncertain as to when you will gain control, which makes it hard to make some of the split-second reactions that are necessary with some of the given starting points.
To load a save state later, just choose the game off the main menu, enter the game, and then at any time while the game is running (even if you’re on the title screen or something), open the system menu and select “Load.”
Games that support passwords continue to allow you to utilize passwords. You use them exactly as you would in the NES versions of the games.