MTG: Fiend Binder and wider implications of Rule 508.1d

If there is a cost for attacking, the player doesn't have to pay that cost and can proceed anyway? What's the point then?

First, that's not relevant, as this is not a cost. Costs are things that must be paid to do something, not something that happens as a result of something. (If a card says "When this enters the battlefield, sacrifice a land", for instance, then sacrificing a land is not, for rule purposes, a cost, even though the card "costs" you a land in the plain English sense.)

Second, this doesn't mean you can attack with a creature even if you don't pay a cost required for it to attack. It means you can refuse to pay the cost and then not attack. For example, Trove of Temptation has the ability "Each opponent must attack you or a planeswalker you control with at least one creature each combat if able." Propaganda says "Creatures can't attack you unless their controller pays 2 for each creature they control that's attacking you." If you have both Trove of Temptation and Propaganda on the battlefield, your opponent can refuse to pay 2 and not attack you with any creatures, even though that means that they are unable to satisfy Trove of Temptation's requirement.

the rule doesn't specify "mana cost" but I presume that is what it means, as opposed to a requirement, like "sacrifice a creature"?

Sacrificing a creature can be a cost. It's important to distinguish between when sacrificing a creature is a cost or an effect, since if it's a cost, you generally can refuse to do it (though you then don't get the effect that it's a cost for).

Fiend Binder's ability is not a combat requirement or restriction. It is a triggered ability that happens as a consequence of attacking. You can attack with it if your opponent controls only tapped creatures, or even no creatures at all.

Your question has a number of incorrect assumptions:

First If a creature can't attack because of one of its abilities, it's called a restriction, not a requirement:

508.1c The active player checks each creature they control to see whether it’s affected by any restrictions (effects that say a creature can’t attack, or that it can’t attack unless some condition is met). If any restrictions are being disobeyed, the declaration of attackers is illegal.

A requirement is given when an effect says that a creature must attack:

508.1d The active player checks each creature they control to see whether it’s affected by any requirements (effects that say a creature attacks if able, or that it attacks if some condition is met). [..]

Second Since Fiend Binder's ability is neither a restriction nor a requirement, its ability can't be a cost for attacking or defending either.

Third Fiend Binder's ability is a triggered ability. It goes on the stack whenever its trigger event happens.

603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”

Whether a triggered ability can't go on the stack, is countered, or resolves without taking effect has no influence whatsoever on the event that triggered it, because that event has already happened, it's in the past.

A few things can happen between the triggering event and the ability actually doing something. A triggered ability might not go on the stack at all, because an effect says so (e.g. Hushbringer), or it's immediately removed because it lacks a valid target:

603.3d The remainder of the process for putting a triggered ability on the stack is identical to the process for casting a spell listed in rules 601.2c–d. If a choice is required when the triggered ability goes on the stack but no legal choices can be made for it, or if a rule or a continuous effect otherwise makes the ability illegal, the ability is simply removed from the stack.

This would apply to Fiend Binder if your opponent controls no creatures, or none of them could be targeted by Fiend Binder.

The ability could be countered, e.g. Disallow).

Finally, if Fiend Binder targets an already tapped creature, it will resolve but nothing happens, because, as you correctly note, a tapped creature can't be tapped. However, this is inconsequential, since in this case, tapping is an effect, not cost.

701.20a To tap a permanent, turn it sideways from an upright position. Only untapped permanents can be tapped.

609.3. If an effect attempts to do something impossible, it does only as much as possible.

So in essence, you may freely attack with Fiend Binder, without any restriction. If you do attack, and your opponent has any creature that you could target with Fiend Binder's ability, you have to do so. If not, nothing happens, and your Fiend Binder keeps attacking.

You're getting combat requirements mixed up with triggered abilities.

Triggered abilities:

603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”

Now, you know that if there isn't any targets, there's a problem, but that's handled by:

603.3d The remainder of the process for putting a triggered ability on the stack is identical to the process for casting a spell listed in rules 601.2c–d. If a choice is required when the triggered ability goes on the stack but no legal choices can be made for it, or if a rule or a continuous effect otherwise makes the ability illegal, the ability is simply removed from the stack.

So, if your opponent has no creatures, the ability triggers, but since there are no legal targets, it's immediately removed from the stack.