Others have taken it but for the sake of knocking out a list
1. That is more likely nxe2god.
nxe is a menu change that the 360 underwent (see blades menu for the older version) and with it came the ability to install a copy of the disc to the hard drive but use the disc for authentication.
god aka games on demand is an early take on downloadable games on consoles and obviously did not need a disc.
Data wise there is no difference between nxe and god so when homebrew came along for the 360 you got the option to convert it.
To that end you can put a disc in, copy it to the drive using the normal menu, run nxe2god and have them appear as god installs on the normal menu and whatever else.
It is far from the only way to install such games or play them -- if you have a homebrew enabled 360 (as opposed to a DVD modded one) then you can convert isos to GOD format (see iso2god), extract isos to folders and run them directly using other homebrew programs from either the internal hard drive or a USB drive, and most will usually fire them across the network via FTP to land it on the internal drive (and probably USB as well if you can't be bothered to unplug and replug).
2.eh...
The 360 had limited backwards compatibility with the original xbox by virtue of an emulator MS provided and put on a separate partition on the hard drive.
Hacked versions of this emulator later happened that tried to allow more to be played, as well as homebrew/hacked versions of original xbox games.
JTAG/RGH 360s can and often do have hard drives replaced but not everybody doing them will install either the emulator as a whole or a hacked version of it. To that end you want to ensure you have a hacked version on there. From there I am not sure what that menu setup wants for original xbox games.
3. I am not sure I would. 360 homebrew has some nice stuff but compared to the original xbox with its amazing collection, wii or anything modern on a raspberry pi...
4. Xbox arcade is more likely a shorthand version of xbox live arcade aka XBLA aka downloadable games that are not full games and are usually smaller games (usually also noted alongside DLC in hacking circles as it uses many of the same formats and has many of the same requirements to run it). There are loads of great XBLA titles -- bomberman act zero is absolutely atrocious, bomberman XBLA is right up there with the best of them, perfect dark zero is an unfortunate casualty in early 360 FPS era but perfect dark XBLA is the best version of original perfect dark you will see (high res, high framerate, all good stuff) until we get some decompiled versions in a few years (see what went for Mario 64 once it got a decompilation), bionic commando main disc game is less than stellar but perhaps not quite as offensive as the other examples where the XBLA effort is a wonderful little title and it is not all just ports and same franchise but the XBLA games in general.
Equally DLC has a lot to offer and has many examples of devs having done the main game, knowing the engine and thus then able to make something quite special. There are limited numbers of DLC titles you can install without modding on a stock console (see PIRS DLC) but this allows you it all. Sure you have horse armour level nonsense, and endless pay us £3 for this track in a music game (guitar hero, rock band, dancing games... were still the toast of the town even if being run into the ground so yeah), but you also get various Borderlands DLC which took a wonderful game and made it better still.
5. 360 homebrew is not really anything special, especially not compared to what came before and afterwards. Most things I would care for are tools to allow you to deal with official games. There are some emulators but I would far sooner grab a raspberry pi or old laptop instead these days.
It has some wonderful games, truly other than PC I think the 360 might have my favourite game library of anything on a big screen, and a JTAG/RGH box will in turn allow you to play games from all regions (as opposed to semi locked -- not all games will be locked, some will be locked to certain regions, abgx360 database then having what each were. Usually it was how much a company feared people importing a game if there were delays between regions but there were many exceptions), install games to USB/internal drive (said internal drive being any that fit as opposed to stock or one of a few select models), play DLC or XBLA, play with cheats, play modded versions of any of those (fair few mods of some merit are out there), possibly redirect controls to keyboard and mouse without need of expensive adapter, play system link* online without worrying about the ping limit they put in**, though in such scenarios I think it was only the host cares so you can happily play between a modded and stock box if the right one hosts).
It would have been nice to have a homebrew scene that rivalled, or indeed by dint of the extra power blew away, the original xbox but we never got it. The wii was contemporary too it and had a wonderful scene of its own so guess it was just difficulty of access (JTAG coming later and having very specific requirements, RGH later still and even more specific and fiddly). Today cheap PCs, android tablets and the likes of the raspberry pi likely mean we will never see such a thing again but hey I am all about doing things rather than having a specific platform so I am not complaining. To that end probably better to content yourself with the commercial offerings and playing them as well as you can, and quite a bit of it will do well on here vs finding it on PC and getting that modded to a suitable level.
*list of system link games
https://www.co-optimus.com/games.php?system=1&lan=true&playerCount=2&countDirection=at least#
**if you were around on the original xbox then all those games that allowed system link people then stuffed over a VPN service (xlinkkai being particularly popular) such that they appeared on the same network as far as it was concerned and play happily without paying for xbox live. Could even be done on an unhacked xbox, though anybody that knew about such things likely had a hacked xbox as there were means to dual boot and still use xbox live if you cared about such things.
