Thank you, it sounds really great!

Thank you! 

1. Is there somebody you could recommend? I'm not sure how much should I trust the reputation of a PI based on web search results only; any confirmation from a real person (especially from this community) would be of great help; 
2. Do you have any idea how much could the services of a PI cost? I'm afraid that we do not have a lot of money, especially by California standards; we had to flee Russia at the start of the war, and Igor's disappearance coincided with us trying to settle in a new place and figuring out new sources of income.

At what point should we contact the PI? I understand that playing to win might mean that we utilize all the resources at once since time is an issue; on the other hand I don't understand the effectiveness of the police + community, so maybe in the early stages of the search when raw data is gathered / awareness is raised PI won't have much to work with.

A Russian folklorist found that about 8 percent of her collection of thousands of lullabies were songs wishing death on babies, presumably weak infants like those mentioned here whose survival was uncertain and who may have been in pain.

I am not sure that this should be taken at face value – death lullabies can be interpreted as a sort of protection ritual, aiming to ward off the actual danger to the child. See, for example, https://hekint.org/2017/01/30/death-lullabies-in-russian-culture/