1. You can do either, really. Most people prefer taking them out in the evening, when it's closer to their natural awake time. Cuddly hedgie might still sleep on you anyway, and more curious/adventurous ones may prefer to explore. Usually you end up waking them up for bonding time, depending on what their personal sleep/wake schedule is. Babies sleep a LOT, so you'll probably have to wake him up. During the day is fine to get them out if you want, you just want to make sure you're letting them sleep and not disturbing them for too long. They may also be grumpier when you wake them up initially.
2. You'll want to keep a lamp on near them to make sure they're getting the full 12-14 hours. The amount of daylight changes with the seasons, and sometimes it can be dark on stormy days and affect the amount of light.
3. You really need a wheel. They're active during the night, and just having them out in the evening isn't going to fulfill their exercise requirement and keep them busy during the night when you're sleeping. If you get one from the store, make sure it's a Comfort Wheel, the largest size (12"). That's really the only safe store-bought wheel. The best wheel you can buy is a Carolina Storm Wheel from LarryT. You can find those and his bucket wheels here - http://www.carolinastormhedgehogs.com/t ... wheel.html
They're completely safe, quiet, and VERY easy to clean. Comfort wheels are safe, but they're a bit noisy, and they're a pain to clean with the ridges.
4. Not preferable. They can cause impactions, and live is healthier and safer. Mealworms are very easy to keep live, you can buy a container, keep it in the fridge, and just take it out once a week to let them warm up and eat. They can last for weeks-months this way.