Couple things, good job on gut loading your crickets before feeding him as it provides extra nutrition to your gecko. As for climbing, that is normal for him to climb up and down things so don't worry about that. With the calcium, make sure when you put your crickets/mealworms (which you can also feed him a couple of those in a deep dish every feeding as they love them) make sure you dust the food you feed your gecko as they won't really just lick calcium from the dish, also provide a vitamin supplement as he/she is still young. Its good you put the moist box in his cage however, you will not need it until he starts turning a grayish white color as that is when you will know he is shedding his skin. From what you said of your gecko being 3-4 inches, I would myself consider that a juvenile/young adult. To sex your gecko, look on the underside by the base of the tail, both will have a V by the tail however if your gecko is a female, they will be hard to see and if it is a male the V will have a purple or brown color to it and also, you should be able to see whether or not he has 2 bulges by the base of the tail. It may be hard for you as you haven't sexed geckos before but if you want to email me with pictures, click on my profile and shoot me an email and I can give you my best opinion on if you have a male or female. Anyhow, about the tail finally, it can be in part to having a bad shed however, even without a moist box, if the pet store was keeping the humidity at a decent percentage than he shouldn't have had many problems shedding. It could also be though that he is younger or even that that is just how his coloring turned out. There are many different morphs (varieties) of leopard geckos now a day. With the fact that he is eating, using the bathroom and is still active, I wouldn't worry about anything.
As for hand taming, just keep what you are doing now, get him used to you slowly and keep stroking him, after a week hold him while your hand is still in the tank in case he gets uncomfortable, then after he seems to do fine with that, hold him outside of the tank but close to something that is soft in case he jumps as you don't want him to get injured. While outside of the tank keep stroking him (as weird as it may seem, talk to him in a calm voice) then also let him climb from one hand to the other. Even set him down on your bed and let him climb around there but keep a close eye on him as they can get out of sight quickly if they want. It will take time, but it seems like he is coming around to you rather quickly :)
Keep doing the great work and best of luck with your juvenile gecko :)
Personal experience with geckos for 2 years