Phylactery Lich’s first ability doesn’t target the artifact. You can choose an artifact that has shroud, for example.

If you control no artifacts as Phylactery Lich enters the battlefield, its first ability won’t do anything. As soon as it enters the battlefield, its last ability will trigger (unless you control some other permanent with a phylactery counter on it) and you’ll have to sacrifice it.

If Phylactery Lich and an artifact are entering the battlefield under your control at the same time, you can’t put a phylactery counter on that artifact. You must choose an artifact you control that’s already on the battlefield.

Phylactery Lich’s last ability is a “state trigger.” Once a state trigger triggers, it won’t trigger again as long as the ability is on the stack. If the ability is countered and the trigger condition is still true, it will immediately trigger again.

Phylactery Lich’s last ability checks whether you control any permanents with phylactery counters on them, not whether you control any artifacts with phylactery counters on them. If an artifact with a phylactery counter on it somehow ceases to be an artifact, Phylactery Lich doesn’t care.

Phylactery Lich’s last ability checks your permanents for any phylactery counters, not just the specific one that it caused you to put on an artifact. For example, say you put a phylactery counter on a Demon’s Horn as Phylactery Lich enters the battlefield. Then you put a phylactery counter on a Crystal Ball as another Phylactery Lich enters the battlefield. Then Crystal Ball is destroyed. Since you still control the Demon’s Horn, the last ability of neither Phylactery Lich triggers.

The game continually checks whether you control a permanent with a phylactery counter on it. The moment you don’t, Phylactery Lich’s last ability triggers. The ability doesn’t check again, so you’ll have to sacrifice Phylactery Lich when it resolves even if you wind up controlling a permanent with a phylactery counter on it by then.

Lethal damage, damage from a source with deathtouch, and effects that say “destroy” won’t cause a creature with indestructible to be put into the graveyard. However, a creature with indestructible can be put into the graveyard for a number of reasons. The most likely reasons Phylactery Lich would be put into a graveyard are if it’s sacrificed (perhaps due to its last ability), or if its toughness is 0 or less.