I wouldn't say a heat lamp "does nothing" for the snake. Being exothermic creatures, raising the actual air temp is overall just as, if not more, beneficial to a snake than relying solely on ground heat. I don't know of any or a lot of actual proven evidence that belly heat aids in digestion. And I know for a fact it's not needed for proper digestion. They just have to get their body to proper temperature, which can be achieved just as easily from warm air as warm ground. Not that I condone heat lamps for everyone, but they are perfectly viable options. I currently use one. And, UTH aren't without their cons too. I currently use one of those too.
As for the OP...
1. Burrowing is perfectly normal. Especially for younger corns and especially when using substrates that are fun and secure to burrow in/through, ie aspen, eco earth, etc.
2. A corn doesn't need an actual light at all. Room lighting will be more than sufficient. Though a light on top or inside (in the case of larger pvc enclosrures, etc) provides more viewing illumination. But that's really more for the owners than the snake. You might find that if it is a normal light bulb in a dome fixture, it will increase the temp and will definitely dry the air. These are things to watch out for and consider. If you really want viewing illumination, I highly recommend LED strips.
3. It's always preferable to let new snakes have a week or so settling in period. This includes minimal handling and no feeding. It helps reduce stress and allows the snake to more quickly and easily feel more comfortable in their new home. Missing a meal for a snake isn't a big deal at all. It's really about the equivalent of an adult human missing lunch. In fact, accept this now. You will more than likely experience at least one and probably more feeding strikes over the years. Many adult snakes can and will go off feed for months, some reaching half a year or longer. You should get in the habit of weighing your snake before each meal though, so you can track weight gain and loss.