Is there a website where I can find the old games that where pre-installed in the Apple IIc ROM? When I could not get an OS disk, I could still have some fun with my Apple IIc by playing those arcade games from the ROM (space-invaders, …) and there was a shortcut that gave me access to the source code, too. Also, is there any corresponding emulator for modern PC/Mac computers?
I don't remember any embedded games, but I would highly recommend Virtual II (http://www.virtualii.com/) for Apple II emulation. Its free if used in a limited fashion, otherwise you need to buy a license, but it does just about everything you could possibly think of.
As far as a short-cut for access to the 'source code', the only thing I can think of is that you are asking for how to access the ROM monitor. From the command prompt:
This will drop you into the monitor, where you can query RAM address and disassemble whatever is in memory (RAM/ROM). For example, after entering the ROM monitor you will get a * prompt. You can type a memory address followed by the letter L (with no spaces) to list (disassemble) memory, like this:
More information regarding using the monitor can be found here: http://www.classiccmp.org/cini/pdf/Apple/Apple%20II%20(Redbook)%20Reference%20Manual%2030th%20Anniversary.pdf starting around page 68 or so.
I feel like you're remembering the Apple At Play disk that came with the Apple //c. It had Lemonade Stand, Quick Quiz, and Space Quarks (more of a Galaxian clone than Space Invaders, but I didn't know Galaxian when I was young, so I compared it to Space Invaders, too).
On Windows 10, I use AppleWin, which is really a //e emulator, but it has all the nostalgia of the //c I loved as a child. I used it to capture the screenshots above. I've never used a Mac, so I can't help you there.
And there used to be an online repository of all things Apple // called asimov.net. There are still mirrors floating around the web. I don't know how well they can be trusted, so I can't specifically recommend any site, but I did use an asimov mirror in order to obtain the pic of Space Quarks.
Also, archive.org lets you play ancient emulated games directly in the browser.