"Wood Outgassing" - What Should I Know?

I would use Cedar for the chest.

I'm a woodworker and I would never recommend cedar for storage of metallic items. This wood is well known for the strong odors it releases over long periods of time. It's great for repelling insects but odor release = damage to metals. Probably the safest wood that is easy to obtain and work with is poplar. Maple is good but much harder to fabricate. Oak, cherry, walnut all contain substances that will seriously stain silver (and other raw metals) as noted above. Virtually all evergreens (pine, spruce, fir etc) will release volatile substances that could harm coins. You might consider building the cabinet of wood, then using separate polypropylene containers for the coins.

Gerstner tool boxes are made of wood and I have never heard any complaints from tool makers who use these.

I was beaten with an answer to that. Note that Cedar is used for moth repellant since it emits gasses for long periods of time. Cedar blocks and hangers are used for clothing places since it keeps away most insects.
If you think about it any wood that you can smell must be giving off some kind of vapors that create that smell.
As to keeping it off the floor flat, legs would help dissapate moisture with increased evaporation due to air movements. Whoever is making this should be instructed to seal everything including the bottom and rears. As I mentioned though, many wood people will tell you that some sealants tend to get sticky in warm areas which is one reason for not sealing the parts of a drawer that must slide. Many of the more expensive furnature makers, such as Thomasville, tend to place metal sliding systems in for the drawers. This makes the drawers always slide and then all sides of a drawer can be sealed.
Any type of moisture absorbing material is great except most people tend to forget that they do not absorb moisture forever. Eventually they get what could be called full. Then they may as well be a lump of clay. Only use them if you know you can remember to change them.
One little suggestion is if it were me, I'd have a sort of left out tray or something for the top section. It appears to be excessively deep for no reason. Possibly a half shelf on one side for small firearms and leaving enough room for rifles and/or shotguns to slid into place. Such a shelf would leave all drawers free for coins. Or have more drawers for coins. Or make those present drawers deep enough for a left out tray so you could put in more coins.
Another idea is to check with coin dealers as to where they get those trays they use at coin shows that have places for individual 2x2's.
Hope I'm not boring everyone, as usual.
OOPps. Had to edit this to add one more thing. As to a lock on that cabinet. Great if you have kids around but don't bother with a large horrible one since anything wood is easily opened with a crow bar.

line the inside of the coin area with stainless steel sheet, also have a hidden drawer or compartment to put the extra specials in.

Now here is an interesting thought. Have that made a lot thinner, higher legs and make the top sort of a marbly or waterproof type of material. Now you have a coffee table with a concealed gun compartment. Next have two smaller and similar ones made for end tables only now with concealed drawers for coins.
You now have custom made furniture suitable for a living room.
You now have a safe place for guns that is almost a no look for by thieves.
You now have no problems with guns and coins in the same cabinet.
You now have a much larger area for coins and the ability to sort as you like.

Hang on...adding the word desiccate to my vocabulary.

@just_carl: I like the way you think. In ten years when the technology becomes more mainstream I'll take your idea to the next level. Sometimes we have projects that use "smart glass" (http://en.wikipedia.org/wiki/Smart_glass). Right now it's mostly electrochromic - apply a burst of electricity to switch from clear to opaque glass. When you're home you have a coffee table display case. When you leave the house just flip a switch and everything becomes hidden behind the colored glass.

If security is secondary to just keeping kids out, use a pin lock. Drill all drawers from top to bottom, drop a long metal rod through the holes. Nothing pulls out until you pull the rod straight up.