The mana produced by Karn's last ability can be spent on anything that isn't a nonartifact spell. This includes casting artifact spells, paying costs to activate abilities of both artifact and nonartifact permanents, paying ward costs, and so on.

Karn's last ability isn't a mana ability, even though it adds mana. It uses the stack and it can be responded to. Use the number of artifacts you control as the ability resolves to determine how much mana to add.

The ability that defines Karn's power and toughness works in all zones, not just the battlefield. As long as Karn is under your control and still an artifact, its own mana value will count. In most cases, it'll be at least 5/5.