Plague Engineer doesn't have anything to do with counters, so there's no interaction whatsoever with Solemnity. Only if a spell or ability explicitly mentions counters does it use counters (in the card text or keyword rules). What the Engineer has is a static ability generating a continuous effect, which lasts only for as long as the Engineer is on the battlefield and has the ability. So it starts immediately with it entering the battlefield and ends immediately when it leaves or loses the ability.
I am thinking no.
And lastly, is Plague Engineer's effect dependent on ETB? Again, I am thinking it is not.
P.S. Sorry I goofed the name of the card. Fixed it in thread.
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