If an effect begins to apply to the target artifact before it becomes a copy of another permanent, that effect will continue to apply.

Saheeli's first ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

If the target artifact is an Equipment attached to a creature and it becomes a copy of a non-Equipment permanent, it becomes unattached. If it becomes a copy of an Equipment, it remains attached.

The target artifact is an artifact in addition to whatever types the second target has, and this exception is copiable. If something else copies the artifact later in the turn, that copy also will be an artifact.

Saheeli's loyalty ability causes the target artifact to copy the printed values of the target permanent, plus any copy effects that have been applied to it. It won't copy counters on that permanent or effects that have changed its power, toughness, types, color, or so on. Notably, it won't copy effects that made the target permanent become a creature.

If the target artifact copies a permanent that's copying something else, it will become whatever the target is copying.

If the target artifact isn't an Equipment and becomes a copy of an Equipment, it'll become unattached when it becomes a non-Equipment artifact again.

If the target artifact becomes a creature the same turn it enters the battlefield, you can't attack with it or use any of its abilities unless it has haste.