Welcome to GF! Congratulations on your new gecko. Here are my answers to your questions:
1. Do I need a calcium dish? And if so, what exactly do I purchase from the store?
The prevailing wisdom these days is to dust the feeders every other feeding with calcium and vitamin D3 (I use Repashy Calcium Plus) and to keep the calcium dish out of the enclosure. Although some people feel strongly that the gecko knows exactly how much calcium it needs, other people feel that the gecko can easily get too much calcium and end up with symptoms (oddly enough) that look somewhat like those of a gecko that hasn't gotten enough calcium. I have been keeping the calcium dish out of the gecko cages (except for hatchlings) since 2009 and it's been working well.
2. Should I be concerned he isn't ever coming out to explore yet?
It's very normal for a new gecko to spend nearly all its time in the hide. Some of them, unfortunately for us, seem to want to spend all its time in the hide indefinitely. I would recommend giving it a week without handling it. Then, in my opinion (others may disagree), I feel it's OK to take it out of its hide maybe once a day. Some geckos seem to like being handled (or at least like the warm hand), others tolerate it, and a few actively dislike it
3. Is it safe to use the infrared heat lamp, or should I strictly rely on the heat mat?
In my opinion, it's easy to get the air in the enclosure too hot. I feel that if the ambient temperature is such that humans are comfortable, it's fine for the gecko. Even in the winter when it may be in the mid-60's in my house, if the gecko is cold, it can go into its warm hide and the ambient temp will be warmer because of the trapped warm air in there. When we heat the enclosures so that the air temp is constantly above the low 70's, I think it's possible to dehydrate them. Here's an article written for Gecko Time by my vet about this: http://www.geckotime.com/eyelid-dyse...eopard-geckos/
4. What is the best way to handle him, if he isn't coming out from hiding ever?
See above. Also, geckos tend to be more comfortable if you make a little cave for them with your hands as opposed to holding them around the body because they feel less like prey that way (in the cave).
Feel free to ask other questions. Always glad to help.