Why spreads exist even without costs: a Bayesian market maker sets the ask and the bid as the expected value after a buy and after a sell.
The Glosten–Milgrom model explains the bid-ask spread with information alone. The asset is worth either a high or a low value, and the market maker does not know which. Each arriving trader is informed with probability π and knows the true value, or is a noise trader who buys or sells at random. A buy is therefore a hint that the value is high. A competitive market maker who breaks even on average sets the ask equal to the expected value given a buy, ask = E[V | buy], and the bid equal to the expected value given a sell, bid = E[V | sell]. The spread is the gap between those two conditional expectations.
Because a buy order carries information. If some buyers know the value is high, the market maker's best guess of the value after seeing a buy is higher than after seeing a sell. Setting each quote to that conditional expectation means the maker breaks even against the mix of traders, and the gap between the two is the spread. More informed traders, a larger π, means a wider spread.
By Bayes' rule. After every trade the maker updates the probability that the value is high: a buy raises it and a sell lowers it, by an amount that depends on π. Over many trades the belief drifts towards the truth, the quotes follow it and the spread narrows as uncertainty shrinks. The lab draws this belief path trade by trade so you can see the learning happen.
Yes. Given everything seen so far, the expected next belief equals the current belief, so the quoted mid has no predictable drift. Beliefs do move, sometimes a lot, but never in a direction the maker could have forecast. This is the same idea as an efficient price: the price already reflects all public information, and only new trades move it.
The noise traders. Against informed traders the maker loses on every trade, against noise traders it gains the half-spread, and the quotes are set so the two cancel on average. A common misconception is that the market maker profits from the spread here; in this model its expected profit is zero, and the spread is a transfer from uninformed to informed traders.