Does redrawing a card reset the fatigue count

The key concept at work here is that of Card Advantage. The basic premise is that cards are extremely valuable, and having access to more of them than your opponent is a strong advantage.

Perhaps counterintuitively, card advantage is more important than mana advantage (especially in Hearthstone where mana is pretty normalized), and card advantage is more important than life advantage (especially in the early game, when a few life points doesn't affect anything). If you think about it, one minion card can be worth 10 life points if it gets some good attacks in. A spell card can also be worth a lot of life, if it kills another creature which can potentially wreck you.

Framing your examples using card advantage:

Elven Archer

• Option 1: Play on first turn, then get steamrolled by a 2/2 or killed by damage. Net effect: Gain 1-2 life advantage, but lose 1 card advantage.
• Option 2: Play when it can kill an X/1, then maybe even attack or block before dying. Net effect: Break even on cards (though note you may have sniped a far better card than your 1-drop), as well as maybe gain some life advantage.

Loot Hoarder

• Option 1: Burn the coin to play turn 1, then die to a Rogue. Net effect: Break even on cards (due to the deathrattle draw), 2 damage to the Rogue.
• Option 2: Save for later, try to block an X/2 or sneak in an attack. Net effect: potentially up one card (you lose the hoarder, draw one, but you killed their minion and/or made them burn a card) and some life. Worst case is the same as option 1. And you keep the coin!

Now of course there are exceptions, and there are aggro decks that eschew long-term card advantage in favor of agression. But the theory there is they are setting up a "ticking clock" that the opponent is forced to deal with, probably by making non-ideal decisions as far as his/her card advantage.

In general, the main idea is: Cards are powerful resources, far more than a few points of mana or damage. Treat them as a scarce currency, and spend them wisely to get as much bang for your buck as possible.

It says

Look at the top tree cards... Draw one...

Having 2 cards left, you would look at the 2 of them and pick which one to draw and discard the other, no fatigue taken.

Having 1 card left, you would look at that card and draw it, being the lone card left, there's no option for you to choose from. No fatigue taken either (as Studoku confirms in his answer).

Having no card left, you don't take a fatigue hit, since you can't look up at any cards, you can't draw one, no fatigue.

As you can see from the screenshot of my testing, my health haven't changed after playing the second one with no more cards.

Looking at cards doesn't count as drawing them, neither is discarding them considered a draw attempt.

Note that fatigue is represented as a card but it's not a card at all, it's just the way Blizzard decided to represent it. It could have been a stabbing dagger, a chicken biting you or whatever... so don't mistake the look fatigue got with it being anything but a effect of having no more card and having to draw one.