A lot of questions here. Might be helpful to split them into different posts, but let's take a crack at them.
The reason why the second price sealed auction is set up this way is simple. Imagine you have a set of individuals/bidders, thinking of what $b_i$ to bid. They all have independent values, $v_i$ of a single object up for auction (their enjoyment does not depend on others). The optimal strategy in a second price auction is to bid however much you value that object, that is,
$$b_i = v_i$$
Since you won't pay as much as you bid if you win, you have a chance to get a positive benefit from the auction, or if someone bids the same as you, you also get a payoff of zero, which is an, "oh well", but no big deal.
If you unilaterally change your bid so that $b_i > v_i$, then you increase the chance of winning, but if you're still the top bid, then you still pay the same amount, so it seems kinda useless. If you happen to beat out someone's bid, $b_j$, where $b_i > b_j > v_i$ then you win! But now your payoff is negative. You can think similarly for the case of underbidding. It decreases the chance of winning AND it doesn't increase the chance of paying less for your object, so that's two bad things.
Basically, in theory, second price auctions induce truthful bidding.
Compare that to the first price auction. If you simply bid $v_i = b_i$, then your expected payoff is exactly zero. Very unfun. So instead you "underbid" your "true value". It decreases the chance of winning, but if you do, then you actually have a positive payout. The amount you underbid is obviously based on what you think other people's values are. So bidding here in first price auctions is strategic, rather than truthful.
But the big result you should know is that, in theory:
First price and second price auctions are revenue equivalent for the auctioneer, given independent private values for a single object without budget constraints.
In theory. But in real life, people's bids are interdependent. You can imagine a bunch of big-wig types in a room for an auction of a priceless piece of art, where part of the value of the object is winning it and beating out the competition because you're the coolest kid on the block (with lots of cash to throw around). In this case, you can imagine a first price auction would be better suited for the house than a second price sealed auction.
