DnD 4e was balanced around the idea that monsters have an AC around level+14, other defenses around level+12, attacks against AC around level+5, against other defenses level+3. There are slight deviations, Solos have more of everything, Soldiers have more defenses, Brutes have more Fort, less Reflex and attack, but these are the approximate numbers.
On the other hand, PCs should have attacks and defenses along the same numbers. They gain attack bonus equal to half their levels, +Attributes, +Expertise, +Proficency (if Weapon attacks) and +Enchantment bonus from the magic items. The same is true for AC and other defenses.
For this reason it is essential that they have the Enchantment bonus they should, otherwise the level appropriate challenge (according to the DMG) will kill them easily. This is not so big a difference in low levels, no magic or +1 is not a big deal, but +6 or +2 could mean life or death at level 27.
This means magic items are not a story element like in earlier editions, but as an integral part of a character as feats for example. So the question is not whether the Dwarf Barbarian should get a magic hand item, but rather why does he not already have a neck slot item, and how high is the bonus of his Weapon and Armor.
Some people, mostly those with roots in DnD 2nd edition, are not comfortable handing out magic items like candy at Halloween, and think magic items are a gift and something special.
If you are one of them, you should consider introducing the inherent bonuses as described in DMG2 and perfected in the Dark Sun setting book. It takes care of the numbers, and you can keep magic items rare and interesting. It is also a good guide to see what kind of bonuses PCs must have from Enchantment.
If you do not have access to any of these books, just remember to upgrade every main item (weapon/implement, armor, neck slot) after level 1, 6, 11, 16, 21, 26. If this is taken care of, you can give other magic items.
From Player's Basic pg 76
Most DMs have a monster die the instant it drops to
0 hit points, rather than having it fall unconscious and
make death saving throws.
Mighty villains and special nonplayer characters
are common exceptions; the DM might have them
fall unconscious and follow the same rules as
player characters.
Also, next section:
Sometimes an attacker wants to incapacitate a foe,
rather than deal a killing blow. When an attacker
reduces a creature to 0 hit points with a melee attack,
the attacker can knock the creature out. The attacker
can make this choice the instant the damage is dealt.
The creature falls unconscious and is stable
So in general enemies die when you drop them to 0. Occasionally, a DM might want a villain that has the potential to pop back up and thus give them death saves (I'd recommend DMs tip this off somehow).
If you have need of an enemy after they are out, you can choose to drop them to 0 and have them be unconscious (technically, this only works with melee attacks, but really there's no reason to not allow it to work with all forms of attack).
Best Answer
There doesn't appear to be a hard-and-fast set of rules in 5e about what constitutes "art object" vs totem or focus. Your best guideline is likely the DMG, specifically Chapter 7: Treasure. The chapter opening divides possible treasure into four categories: Coin, Gems, Art Objects, and Magic Items--the separation is important, implying that Art Objects have no inherent magic, or magical properties, of their own.
Taking a look at the treasure tables on pp. 133-134 backs this up; the available ideas for Art Objects are all items of visible worth (usually involving precious metals, gems, and fine workmanship) but never make mention of any magical capability at all. So it appears that items of fine craftsmanship, without magic to them, can be considered Art Objects and thus retain their value at sale, but items with magical properties cannot--they become Magic Items.
It does appear that items need to hit a certain level of aesthetic expense and craftsmanship in order to qualify, though that may be just the limitation of the reference here. So expect a DM ruling about the relative value of your wooden statuette, presumably taking the quality of carving and the rarity of material into consideration. Rough-hewn abstract carvings made of the region's most common tree probably won't do as well as a meticulously detailed piece of a rare wood from another land.
So, moving on to your second question: when evaluating totems or foci, that leaves us with two main metrics: quality of craftsmanship, and whether they're considered magic or not.
The question of whether basic foci, regardless of class, have enough innate magic to be considered "magic items" in the first place should probably be another question on its own. I would argue it's up to the DM to make a ruling here one way or the other--either a basic focus has some small amount of innate magic in order to enable it to be used as a focus in the first place, or it doesn't, since it can be found on the basic Adventuring Gear table, which has no magic items on it at all. Whichever argument you go with will inform your game and your world, even if only in flavor.
A judgment on the quality of craftsmanship also belongs to the DM here. As druids generally eschew a lot of the high-expense materials found in art objects, druidic totems are subject to the same criteria as your wooden statue: craftsmanship, and innate magical properties. You could make more of an argument for a Holy Symbol of an amulet or reliquary to be "blingy" enough to be considered an art object, considering the long, real-world history of people spending a lot on religious icons to prove their piety.
TL;DR, Art Objects need to match two criteria: high level of craftsmanship/expensive materials, and not being magic. Consult your DM for specific rulings on a particular piece.