If you don't currently have a character unlocked, conversing with them in Adventure Fields, then obtaining an Emblem afterwards, will set a flag to have that character unlocked without needing to do so later on when you see them in the selected character's campaign.
While there's two versions of the track "Choose Your Buddy!", Finger Bass and Slap Bass version, on the OST "Sonic Adventure Digi-Log Conversation", an unused version of the song remains in the sound test of the game, it being the track without a lead guitar, never on the OST.
In Casinopolis, you can enter the pinball machines whilst reading a hint monitor when using a second controller to have Tails enter the machine, of which pulls Sonic in. You can obtain a time of only a few seconds with this, as the timer won't fix itself unless you read it again.
When Knuckles is too far out of a map's field, he will automatically glide back to the main area. In some sections, Knuckles can get stuck gliding back by rotating into another invisible wall that would send him back. If done right, Knuckles will softlock and fly up forever.
According to a Dreamcast 1999 presskit showing multiple early screenshots of the development of Sonic Adventure, there was originally a sign on the building for the hotel entrance, likely to recognize it easier; this was later scrapped in all published versions of the game.
When finishing Windy Valley in Gamma's story, after the cutscene with Angel Island opening up, rather than spawning outside of Windy Valley as you should, the Japanese version (left) spawns you outside of the train station. This was fixed in future versions of the game (right).
While cutscenes in Sonic Adventure on Dreamcast are unskippable, in Sonic's Story, the cutscenes for unlocking Tails and for unlocking Amy can both be skipped by soft resetting the console (A, B, Y, X, Start); the game will continue without any issues with them both unlocked.
While Knuckles' Lost World contains the rest of Sonic's object set outside of his regular boundaries, what's also outside for an odd reason (above the boulder section) is the colored lights for the pressure plates to open the door at the end of Act 1 while never having any use.
Due to the fact DX's Chao code was recycled from Sonic Adventure 2 Battle's as Battle was released first for GameCube, the game files contain leftover textures with drawings made of Dark Story characters that Chao would draw when having a high bond with any of the three.
Update: It seems that this can indeed be done on Dreamcast; my assumption was based on my personal attempt(s) to recreate it, as I had not managed to achieve it successfully. Apologizes for the misinformation.
[Volume warning]
If you cast your line as Big and fall mid-cast, the game has no way of telling where to throw your line, resulting in the game crashing. This is only possible on DX. [Footage by pokemonfan4000 via YouTube)
@CrotchtasticFox It seems that performing this on Dreamcast will act nothing out of the ordinary, having Big react as if you're casting too close, to which he merely whips his rod forward and nothing more. DX seems to have removed some lines of code that used such a fail safe, causing this crash.
@AppuruGumi By choosing the Chao icon and clicking the dropdown section in the center, You'll be brought to a section with Puyo Pop, Chu Chu Rocket, and the Tiny Chao Garden. Selecting it will let you choose which Chao you'd like to transfer to your GBA, which you then transfer it to SADX.
Obtaining a Tails, Amy, and Knuckles Chao are possible without hacking on the GameCube version of DX. You can obtain all 3 with the use of the "Interactive Disc Catalog Summer 2003" demo disc (Japan only), and the Tails Chao is obtainable via Phantasy Star Online: Ep I & II/Plus.