I would want to know if there is tool that would let me check the build date of executable and if it's signed for debug or retail.
No, there isn't. PlayStation 2 binaries are all in plaintext, unless they are protected by either MagicGate or DNAS. Or even both.
Additionally all you can say about a PS2 binary is if it got debug symbols or not. Surprisingly a LOT of games actually shipped with non stripped builds of the executable, wasting memory and leaving precious information about the system behind. One can say that the files timestamp is the only reliable way to reasonably infer about the build date of a game. And only trust that if you extracted it from a silver disc yourself as stuff floating around the internet is often tampered with...
It depends. Having debug information does not necessarily mean that it will take up more RAM because the PS2 kernel will ignore the non-loadable segments. However, it may take up more RAM and be slower if the code was not compiled with optimization, which might be done to make debugging easier.
