I assumed along with your Hygromagnetic Handwavium-Phlebotium(TM) you had some patent-pending SuperRock-Handwavium to make this thing not collapse in on itself and to plug any cracks into the Earth's mantle. So, my best ideas for your questions:
Overall, this canyon would be a mostly dark scar on the landscape with extreme weather near the bottom but few effects on the rest of the world, a normal ecosystem near the top and underwater but with nothing living in the middle.
-How would the canyon look from space: If the sides are very steep, then it would look like the Grand Canyon, except more than twice as wide as the Grand Canyon's widest point. If the sides were more sloped, then it might look more like some of the canyons in the Himalayas. However, the biggest difference in appearance is that this thing would be dark trench in the landscape. At ~30km deep, sunlight would touch the bottom of the canyon rarely. The Grand Canyon is less than 2km deep, and even at that depth the parts along the river get a good amount of shade. Sunlight definitely will reach the bottom for at least part of the day, however much of it will only get a brief glimpse of sunlight reflecting tens of kilometers up the side of the canyon.
-What would be the climate inside of it: Temperature throughout the canyon is tricky. As altitude increases temperature decreases because there is less air to hold the heat(snowy mountaintops), so it makes sense that temperature would be high in the canyon since the air pressure is higher. This gives a rough estimate of 6.5c per km of altitude change, however the lack of sunlight can negate much of that. So, depending on the amount of sun the canyon will have areas that are boiling hot and areas that are freezing cold. According to calculations that use too many letters, pressure at 30km below sea level would be about 15 atmospheres, where 1 atmosphere is sea level, if it was a comfortable 15 degrees Celsius. Pressure can get closer to 25atm in the colder parts, or 7atm in the hot parts. This will cause extremely chaotic storms where these pressure differences meet.
-Effect on world climate: Thankfully, probably small. Wind from the Great Canyon Storms will definitely blow out of the canyon, but most of the churning atmosphere will be contained down below. A 60km wide, 30km deep, 500km long canyon would be 900,000km of air missing from the atmosphere, however the pressure will mean that even more than that would fall into the canyon. The exact amount can probably be calculated, however the pressure differences from depth and temperature results in an equation beyond me, so let's assume this canyon sucks in 5-10 million cubic kilometers of air as a very, very rough guess. There is roughly 4 billion cubic kilometers of air, so this trench will bring the edge of space closer to the earth but thankfully air pressure at sea level will remain basically the same.
-Canyon Critters: 0-3km down: Much the same as the Grand Canyon. Birds, adventurous goats, and very adventurous people, as well as hardy trees, bushes, moss, etc. 3km-5km: In areas that are warm enough to ease the atmosphere pressure, but shady enough to not cause the critters to boil, will likely also look like the Grand Canyon. 5km-'sea level': Extremely high pressures, extreme temperatures, extreme storms... nothing but hardy bacteria until the bottom. Underwater: Plenty of life. I'd imagine the sea would look much like the bottom of the actual oceans: tough critters huddled around hydrothermal vents. Since the bottom of this canyon will be like 5 feet from the Earth's mantle, I'd assume these vents will be plentiful. Water is really hard to compress and air is light, so the pressure here is likely less than what it actually is at the real bottom of the ocean.