Why is the Market-Clearing Price also an Equilibrium Price? When the price is such that supply equals demand for e.g. bicycles, everybody who wants to buy a bike at that price gets one, and every bike gets sold. Everybody goes home happy and the marketplace clears. Hence the price at which the supply and demand curves cross is the Market-Clearing Price.

This isn't enough to make it an equilibrium price. "Equilibrium" means that if the price is there, it'll stay there. If it's a stable equilibrium, then even if it starts above or below that price, it'll get pushed towards it.

I feel like I understand why prices starting below the market clearing price, will get pushed up. Of course a firm will raise prices: they can charge more per unit, and sell more units (assuming a rising supply curve).

If you start above market-clearing price, there is a trade-off, because sales are demand-limited (assuming the demand curve is decreasing). Raised prices would mean more revenue per sale, but fewer sales. Lowering the price would mean more sales, but less revenue per sale. Intro textbooks (e.g. Mankiw) seem to claim that the mere fact of there being a trade-off at all, is sufficient to push prices down, but it doesn't seem like that to me. Whether lowering prices would increase profits depends on the slope of the demand curve; if it's shallow enough, then the large increase in revenue-per-sale could make up for the small decrease in number of sales.

If I'm not wrong then a firm with a monopoly on bicycles would set prices to maximize revenue, i.e. the area of a rectangle whose length is quantity and whose height is price. The corner of this rectangle would lie somewhere on the demand curve, at or above the market-clearing price. The profit-maximizing price might be one where there's excess potential supply (bikes you would have been willing to make and sell - presumably you didn't actually bother making them). 

I assume that competition is what makes the difference, and somehow forces prices down to (but not below) the market-clearing price, making that the equilibrium for competitive markets. But I don't get how this works, e.g. why underselling won't stop being worth it until there's excess demand.
(This is referring both to the initial question and to the comment. I would respond to each individually, but I don't think I have the reputation to address the comment directly.)

Paraphrasing a bit from your question, you're right- an equilibrium price is one where no firm has an incentive to deviate in their pricing or production decisions, and demand is perfectly satisfied (again, given the price). While the exact "mechanism" that would push price down might differ depending on what market type you're considering, the common feature will be that there is always a marginal incentive to deviate down to the lower price.

In terms of perfect competition- as you mentioned, it can be seen as a race for different firms to undercut their competitors, until they hit the no profit condition. That pushes the market clearing price down to marginal cost, which is the equilibrium price.

For monopolists, they're looking to maximize profit, not necessarily revenue. But, as every firm does, they choose a price p to satisfy:

$$
max\{q(p-c)\} = \pi
$$

They will choose a price higher than the one found under a perfectly competitive framework; hence why the market clearing price, and equilibrium price, is higher under monopoly than perfect competition.

But if the monopolist were to set a price different from the one that satisfies that maximization problem, they are pricing, as you described, too high. That tells them when to produce more units at a lower price; until their profit maximization price is hit. That becomes the clearing price, and the equilibrium price for a monopolist.

I'm sorry if that was too repetitive or elementary. You're question is an excellent one, and a somewhat subtle point has to be made. If it wasn't clear, or you'd like it re-framed, let me know!