Most would not bother to repair a DVD drive save for the little rubber cable (open it, clean it, put it back together) and maybe a laser replacement if you are especially bored.
Power supply. I am not sure what official/semi official replacements go for these days.
http://secure.llamma.com/catalog/xbox-major-parts-c-64_22.html has them save for the 1.6 models down for about $20 USD. What installation will be varies wildly depending upon where you go and what they think, I tend not to offer such services outside my circles of friends (in which case feed me is the payment I seek) and I have all the relevant gear if I have issues so I am not sure what the going rates are. I would not be surprised if it is more economical to get another xbox and mod it. Also though it tends to be OK there is always the chance the power supply decided to take other things out when it went.
Reliability. Without question the original xbox is more reliable though the later 360 models mostly sorted their issues.
360 softmodding does not exist and the basic DVD mod that most do will not get you far on the original xbox front -- you get to buy chips and fiddle if you want to do anything with the original xbox (I doubt you will find many JTAG machines these days so look up the RGH mod, it will cost you xbox live mind). Likewise the orignal xbox backwards compatibility is not the best, it can get a lot done but there is also a lot missing. Wikipedia does seem to be a source for information this day
http://en.wikipedia.org/wiki/List_of_Xbox_games_compatible_with_Xbox_360
The hacked versions of the xbox emulator will allow for any game to be given a punt but it is not much better.