Hey there! I'm new to the corn snake world too, but when I first got my girl I learned quite a bit. My girl actually burrows pretty often, especially after I exchange her substrate... she loves to swim through it! It's a natural behavior to try to hide. I recommend getting a full size cave or hide for the warm side and an additional one for the cool side. This allows them to chose what temperature they want to be in while still feeling safe. I use a UTH but she never burrows on top of it.. then again, it doesn't get very warm either. During the winter it gets very cold here, so we use an overhead lamp with a very low wattage to keep the gradients in the proper range. We use a screen lid, with the lamp on top, that way the snake can't get to it. Whichever bulb you choose, pick one with full spectrum uva and uvb light. They can get pricey to replace, but they provide the kind of light your snake would have in nature. then you can use whatever heat bulb you wanted. As far as activity goes, some snakes are just more active than others. Mine will roam around every once in a while when we shut the lights off, but other than that she likes to burrow or hide. Snakes will feel cold to our touch because our body temp is 98, whereas theirs is in the 70s to 80s, making them feel cold in comparison, so that is also normal. They are more active at night, so you may not see it drinking, but they do. The water tends to evaporate quickly in my viv, so keep an eye on it. If you're concerned hold it over the dish (but not in it) and it may bend over to drink or examine the water. It's cute to watch them gulp. Just be patient and you'll soon see what's "normal" behavior for your little one.
Sorry if it's a ramble, but I hope there was atleast a little bit of help in there for you.