Ok, here's some answers:
You can program your exploit in any language you like... So long as you can find a compiler for your code.
PHP is an interpreted language, in that you write the source code and the computer translates it at run time. Most other language don't work that way, you need to compile the code to machine code in order for the computer to be able to execute it.
Now! At present there is no compiler available that creates save games that can be booted as programs. Why? Well, for starters, save games were never supposed to be bootable in the first place; the exploited saves available only work due to glitches in the games used to activate them.
Hence to create such a thing, you need to understand "low level" programming (what you refer to as "binary and stuff"). PHP is what you call a "high level" language - It's very easy for a human to read/write, but far from the actual code the computer ends up running.
So say you learn the machine code specific to the X-Box. Without prior knowledge of assembly and the like (something only a small minority of programmers have), this'll take you quite some time
. Then you can sit down and create an exploit which... already exists! A great achievement indeed.
Ok! So next up, you create your Linux installer. This involves configuring a dashboard to setup a softmod, and then install Linux itself - This could actually be done really easily be editing Krazie's existing installer.
So what's the problem with that? Well, most Linux installers worth using are big. As in, at least a couple of hundred megabytes (intended to be loaded onto a CD). I'm not even sure Microsoft has created any memory cards that big, so odds are anyone who uses your autoinstaller will need to find a compatible USB drive to use instead.
Ok! So say you do all this and create a working autoinstaller for Linux. Well done. Of course, as soon as another version of your chosen Linux build gets released, your work is obsolete.
What you should be doing is studying existing autoinstallers and softmods. Pull 'em apart, see how they work for yourself.
At the moment you're asking questions which, if fully answered, could fill books. It's kinda like explaining to grandma (who's never touched a computer) how to program PHP, if you get my drift... Where do you start?