I've seen this happen before, and it required the xbox be repaired by MS. Halo 2/3 looks really weird in just green. I'm assuming this is one of the older revisions, right? Several years old and no built-in hdmi unless it's an elite? I think you might need a new motherboard sadly. Call around to MS and a local game shop that does repairs to see who has a better price.
Try playing a pristinely-clean audio CD in the console. If that skips in the same way then the issue is the Saturn and not the disc. If it's the console, you might have a dirty lens. Use a lens cleaning disc if you have one, or find instructions on the web for delicately cleaning an optical drive lens. If it's the disc, ask your local used game store if they will polish it for you. Smaller and locally owned stores are more likely to carry titles for older consoles, and more likely to do this for you for free out of nostalgia or if you buy something at the same time.
Ensure that you have a known-good game cartridge inserted before powering on the SNES console. Most older, cartridge-based game consoles don't boot without a game installed. Does the power light to the left of the controller ports turn on when you turn on the power switch? If not, check your power adapter and outlet for functionality and good connection. Check that the reset switch is not stuck. Wiggle it back and forth through its range of motion a few times. If something sticky gets in there it can get jammed in the reset position, which will prevent any game from booting. You mentioned that there is no image. Are you getting any sound from your games after waiting a few minutes with everything connected and turned on? If you hear game sounds or music from the SNES, but see no picture and you've already (really and truly) ruled out the cabling and TV, then you might have a bad main board in the console. If you don't get any video or sound, then it's not booting your game. You can find other...
Put fresh batteries in the controller. If you're using a rechargeable battery pack or rechargeable AAs, try swapping in a new pack or alkaline (normal, non-rechargeable) batteries to see if those are also affected. If the controller works better with alkaline cells, then your rechargeables might be worn out.
Try booting up a game and looking at the screen from different angles. I've read that over time the electrolytic capacitors on the logic board fail (1), resulting in a change in display contrast. Replacing all the capacitors helps in some cases. The only screen adjustment available that I'm aware of is brightness for the backlight, which doesn't affect contrast/viewing angle. Also, take the game gear into a dark room and turn it on. If the backlight is functional, you should see at least some light shining through the screen. If not then you have a backlight issue, possibly in addition to the above issue. Sources 1. Sega Game Gear Repair
The best place to find a network/hdd adapter is at a local game shop. Lots of online retailers gouge, but the going rate at a brick and mortar store is $14.99. Try gamestop as well as locally-owned stores.
First of all, are you using a black and white TV? They made those once a upon a time, and if you're a kid maybe you have parents with an awesome sense of humor. It sounds like you ruled out the game cartridge, so now isolate the issue to the console, cable or TV. Plug your console into a different TV with the same cable to rule out your TV hardware. Plug your console into the original TV using a different cable to rule out your cable. Try a different type of cable (RF, composite, s-video), and if one of them gives color while the others dont, maybe you can work around the problem instead of fixing it. If using an RF adapter, are you using the correct/same channel (3/4) on your TV and the RF box? If you're using an S-video cable, maybe you're only getting luma and not chroma? There should be 4 metal pins and one plastic guide tab in an s-video connector, check for those being bent/broken. If SCART or import hardware... well I hope it's not an NTSC vs. PAL issue. Google for region-specific issues.