Is it legal to use them for making my own mobile game apps, if I alter the pitch and do some remixes? I'm making a 'fan-game' or 'mech-inspired' game, but in an original and legitimate way as possible.
It says right on the page you linked:
These downloads are not public domain, as they are parts of content that has already been licensed and distributed. Although using these downloads may be permissible as long as the project itself falls under the rule of "Fair Use," it is ill-advised to use these downloads for any project intended for profitable gain or commercial advertisement, unless otherwise stated by Kyutwo.com.
15 seconds of any audio can be used for free. I hear this stated on the radio all the time so most sound effects should be fine. I assume you have to credit the original owners in your credits areas though.