Question: Is there a source with records of the participants and/or casualties of the Russian army during the WWI period (1914-1918) and thereabouts? Preferably on-line and indexed/searchable by name(s), birthplace, place of residence and whatever else data the authorities gathered for the recruits.

Answer: 
There is a Russian website that has a project going to provide the data within pdf format spreadsheets. These are also in Russian and provided according to Russian province. You can find these records at <http://svrt.ru/1914/1914-1.htm> .
I've had simliar questions about those records (and those of WWII). For WWI I was directed to try <https://gwar.mil.ru/>