Making Jump skill checks and Tumble skill checks are typically part of a creature's movement, not mandating separate actions
The Jump skill has the entry Action: None. A Jump check is included in your movement, so it is part of a move action and, similarly, the Tumble skill has the entry Action: Not applicable. Tumbling is part of movement, so a Tumble check is part of a move action. A creature that moves on its turn can, during that movement, typically make Jump skill checks and Tumble skill checks during that movement without taking any additional actions.
In the scenario the question describes, the creature has at least two options:
The creature takes a move action to travel some distance to the pit's edge, jumps (probably automatically) 5 ft. in the open space above the pit, and makes a Jump check to jump down (DC 15). Failure means that when the creature reaches the ground the creature is dealt damage and is rendered prone; this tends to end the typical creature's movement. Success by a trained creature (one that possesses at least 1 rank in the skill Jump) means that when the creature reaches the ground the creature treats the fall as 10 ft. shorter therefore possibly being dealt no damage, and the creature lands on its feet. Success by an untrained creature means that the creature treats the fall as 10 ft. shorter and is rendered prone unless the creature beat the Jump skill check's DC by at least 5 (hence, here, a Jump check result of 20 or more).
The creature takes a move action to travel some distance to the pit's edge, moves 5 ft. past that edge therefore falling into the pit, and makes a Tumble check (DC 15). Failure means that the creature is dealt damage and is rendered prone; this tends to end the typical creature's movement. Success means that the creature treats the fall as 10 ft. shorter and, if the fall still deals damage to the creature, the creature's rendered prone.
(The DM doesn't let a creature at the edge of a precipice—or anywhere else—lean over so as to fall into an adjacent space. Even if that would render a creature prone, that's still free movement. For example, a creature with a high enough Tumble skill check to free stand could do that all day without being next to a precipice and never spend any actions to do so. While inchworming across the world this way would be hilarious, the DM doesn't allow even his campaign's most amazing gymnasts to do that!)
After the creature reaches the ground, if it's not prone or through some means can instantaneously stand up from prone, it can continue its move action with any of its remaining speed.
This DM doesn't treat falling as part of a creature's movement—even when done deliberately—as that would see creatures with low movement speeds fall at a slower rate than those with higher movement speeds, and that's silly. Instead, this DM borrows from the rules on Tactical Aerial Movement (Dungeon Master's Guide 20) and has folks fall at a rate of 150 ft. on the first round and 300 ft. each round thereafter. Don't look at this too closely, though, as controversy abounds.
The Dungeon Master's Guide on Falling (303) makes no distinction between making a Jump skill check to jump down and making a Tumble skill check to avoid damage from a fall, but its section on The Environment didn't receive much attention during the 3.5 revision; this reader humbly suggests sticking with the Player's Handbook unless those rules from the Dungeon Master's Guide become necessary.