A creature that regenerates doesn't ever leave the battlefield, so it keeps all equipment, enchantments, artifacts.
Regeneration is a replacement effect which means: "The next time this permanent would be destroyed this turn, it isn't. Instead tap it, remove all damage from it, and remove it from combat."
701.13. Regenerate
701.13a If the effect of a resolving spell or ability regenerates a permanent, it creates a replacement effect that protects the permanent the next time it would be destroyed this turn. In this case, “Regenerate [permanent]” means “The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature, remove it from combat.”
701.13b If the effect of a static ability regenerates a permanent, it replaces destruction with an alternate effect each time that permanent would be destroyed. In this case, “Regenerate [permanent]” means “Instead remove all damage marked on [permanent] and tap it. If it’s an attacking or blocking creature, remove it from combat.”
701.13c Neither activating an ability that creates a regeneration shield nor casting a spell that creates a regeneration shield is the same as regenerating a permanent. Effects that say that a permanent can’t be regenerated don’t prevent such abilities from being activated or such spells from being cast; rather, they prevent regeneration shields from having any effect.
No, except the "tapped" question. That one is kind of right. There seems to be some confusion over what Regenerate actually is. Regeneration replaces a destruction effect so the permanent never dies. It doesn't leave the battlefield so it doesn't lose counters or anything attached to it. It remains the same object.
The "tapped" part is just a function of Regenerate. It doesn't "come into play tapped". It remains on the battlefield and just becomes tapped. Here are the rules for Regenerate:
701.13. Regenerate
701.13a If the effect of a resolving spell or ability regenerates a permanent, it creates a replacement effect that protects the permanent the next time it would be destroyed this turn. In this case, “Regenerate [permanent]” means “The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature, remove it from combat.”
701.13b If the effect of a static ability regenerates a permanent, it replaces destruction with an alternate effect each time that permanent would be destroyed. In this case, “Regenerate [permanent]” means “Instead remove all damage marked on [permanent] and tap it. If it’s an attacking or blocking creature, remove it from combat.”
701.13c Neither activating an ability that creates a regeneration shield nor casting a spell that creates a regeneration shield is the same as regenerating a permanent. Effects that say that a permanent can’t be regenerated don’t prevent such abilities from being activated or such spells from being cast; rather, they prevent regeneration shields from having any effect.
Also when a creature regenerates does it come into play tapped?
Example:
Dutiful Thrull
—Nicol Bolas
Regeneration is a replacement effect which means: "The next time this permanent would be destroyed this turn, it isn't. Instead tap it, remove all damage from it, and remove it from combat."
The "tapped" part is just a function of Regenerate. It doesn't "come into play tapped". It remains on the battlefield and just becomes tapped. Here are the rules for Regenerate:
—Nicol Bolas