It doesn't matter what kind of counters the destroyed artifact had on it, only how many. If an artifact had five fuse counters and two trap counters on it, you put seven +1/+1 counters or seven charge counters on an artifact.

If the target artifact is an illegal target by the time Dismantle tries to resolve, the spell won't resolve. You won't put any counters on an artifact. If the target is legal but not destroyed (most likely because it has indestructible), you do put counters on an artifact.

Dismantle targets only the artifact that will be destroyed. When Dismantle resolves, you choose which type of counters you want and choose an artifact you control to put them on. No player make take actions between the time you make this choice and the time counters are on the artifact.

You can put +1/+1 counters on a noncreature artifact. They won't do anything unless the artifact becomes a creature, at which time they'll add to the creature's power and toughness.