I recently found this Chat Sandbox with the chatbot and wanted to know what all of the chatbot commands do (including learned ones and the ones with no definition.) What exactly was the Chat Sandbox designed for, and what are the commands/what do they do?
The chat sandbox exists on chat.SE and chat.SO. As a general rule, the room with the ID 1 on a given chat site is the sandbox. The one on chat.meta.SE is private (mod room), so the Sandbox on chat.meta.SE is an exception to that. There can also be user-defined sandboxes.
Please note that for participating in chat you need an account with enough reputation to chat (20 reputation on Stack Overflow to use the chat.SO room, or for the chat.SE room, 20 combined reputation on all sites not counting the default 1 for signing up), or an invite from a moderator. The sandbox rooms are not exempt from that rule.
The chat sandbox is exactly what the name says: A chat sandbox. Other rooms may define their own rules for content and actions, where as sandbox is basically "do whatever you want, except abusing flags and violating the 'be nice' policy". This also means it ends up being a room for chatbots, though it's not directly created exclusively for playing with bots (it's also formatting tests, random conversations, starring random stuff, testing chat bugs (recent example), of course chatting, etc.).
As rene mentioned, SE itself doesn't have (public) bots. SE does have some feed accounts (RSS/Atom feeds, posting items as messages in chat, though some come as a ticker instead), but these have negative IDs. If the user ID is under 0, it's an account created by the SE team (examples: feeds, extroverted main man, processed meat). Usually they show up for specific RSS feeds (as far as I know). It's not possible to interact with them though.
All the public bots in chat are created by users. Currently (11.07.18 @ 20:20 +2 GMT), chat.SE only has one bot in the sandbox (disclosure: it's my bot), and chat.SO has 3. In addition, some bots have their own rooms.
TL;DR: the chat sandbox is for testing stuff, such as bugs, bots, formatting, and of course chatting. There's no topic in the chat sandbox (or rules beside be nice, and don't abuse flags), but other rooms can define their own topics and rules (in addition to be nice).
As for specific bots, the available commands can change at any time (and the creator(s) don't need to make it official anywhere), so instead of listing them here (which, IMO, doesn't hold a lot of value since the bots aren't created by Stack Exchange and can be taken down at any point for any reason (or none)), it's generally a good idea to find the source code (if you can). Most bots have documentation (some have more, some have less) on usage somewhere. However, if a specific bot doesn't, you can always browse through the source code.
There are a lot more bots than just those in the sandbox. In addition to the "fun" bots, there are also a bunch of moderation bots (for an instance SmokeDetector, Natty, some SE sites even have their own moderation bots (none of which are made by Stack Exchange)).
TL;DR: the sandbox is for testing chat (within SE rules). The public bots aren't created by Stack Exchange. If you want to find what a bot does, and it's not documented, look up the source code or ask the creator(s).