If I gain control of an equipped creature, do I keep the equipment when it dies?
No.

You don't control the equipment.

Control of the equipment is separate to control of the creature. The same is true for attached auras, or any other kind of permanent. If something gives you control of one, you only control that one unless stated otherwise.

If timing allowed it (e.g. if you took control of the creature permanently, or with an instant-speed equip ability such as Cranial Plating), the equipment's controller could attach it to something else.

However, a lot of equipment cards give abilities to the creature. For example, Hankyu gives the equipped creature the ability to add or remove counters on the Hankyu. You could, for example, tap the creature to waste any counters they built up.

Conversely, Umezawa&#39;s Jitte's ability is on the equipment itself. The controlled creature's attacks would generate counters for the equipment, but the controller of the Jitte would be the one to activate its ability.

If you gained control of the equipment (e.g. Steal Artifact), it would remain attached to their creature, but you could use the equip ability to attach it to something else.