Yes, that's how it works. Solemnity will cause permanents of those types to enter the battlefield with zero counters. In your example Dark Depths' last ability will trigger immediately after it enters the battlefield.
While Solemnity is on the battlefield (C.R. 611.3b, 112.6), no artifact, creature, enchantment, or land (including Dark Depths in normal cases) will have counters on it as it enters the battlefield, since counters being put on a permanent includes counters being given to it as it enters the battlefield (C.R. 121.6). See also this thread.
would it come into play with 0 counters and then trigger the ability that makes marit lage?
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