Why does Steward of Darkshire not buff a Silver Hand Recruit if I have Sword of Justice

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.

Actually counterspell triggers before spellbender in any case.

There is an example here on youtube.

I think it's because in the beginning you cast a spell and then you target a minion. This is not an official explanation though.