The xbox has two main homebrew approaches, three if you count the various versions of Linux made available for it but that does not play here and any scripting afforded by XBMC or something is also not going to be enough.
That being the xbox official SDK, aka the XDK. Various versions of this made available, the last one being an internal one some of the Halo peeps used.
There is also a standard homebrew SDK based on all the usual open source compilers. It was not developed as well for quite a while into the xbox life.
https://github.com/XboxDev/nxdk is probably where I would start today, though I am not sure how developed its 3d libraries are so might want to investigate that.
The XDK was overwhelmingly the most popular during the xbox lifetime. Being made on this is why original xbox homebrew tends not to be downloadable as a simple zip file from a website but hidden in FTP servers, behind walls and on torrent sites like it was a ROM, or only distributed as source code.
As for the specific project of porting a game.
Generally speaking you want either
1) The game to have had source released (be it officially, leaked or recreated)
2) The game to use an engine similar enough to either an existing game or an engine that is open sourced such that you can figure out the differences and port it across.
You can reverse engineer a whole game (be it by playing with a disassembler, decompiler, call viewer and the like or by extracting assets and recreating the physics to match by observing play) but whatever approach you take for something of that vintage and base language (some older stuff is easier, some newer stuff is easier) it is a years long process most of the time even for veteran coders.
Some have kind of gone there as Valve did make and variously authorise certain mods for its Source engine (Source being their own engine a la Unreal, Quake, Doom, ID Tech, Unity and the like are for other companies and time periods).
To the best of my knowledge there are no source releases of any version of Half Life 1. A cancelled Dreamcast port was leaked online but that was a build rather than any source code that might be useful. There was famously a leak during development (and that course of events ultimate gave us Steam) for 2 but it is way different to final.
https://combineoverwiki.net/wiki/Half-Life_2_leak
As mentioned above Half Life has gone on to be remade in newer engines, Black Mesa being the most notable of those. The original xbox quite notably had a version of Half Life 2 made available (fairly stripped back as these things go) so some might consider that approach here.
