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 are some sources but they are fragmented and most of them are not digitized yet.

One online source about WWI casualties for 1914-1915 years is in the [Russian State Library](http://sigla.rsl.ru/table.jsp?f=1016&t=3&v0=%D0%98%D0%BC%D0%B5%D0%BD%D0%BD%D0%BE%D0%B9%20%D1%81%D0%BF%D0%B8%D1%81%D0%BE%D0%BA%20%D1%83%D0%B1%D0%B8%D1%82%D1%8B%D0%BC,%20%D1%80%D0%B0%D0%BD%D0%B5%D0%BD%D1%8B%D0%BC%20%D0%B8%20%D0%B1%D0%B5%D0%B7%20%D0%B2%D0%B5%D1%81%D1%82%D0%B8%20%D0%BF%D1%80%D0%BE%D0%BF%D0%B0%D0%B2%D1%88%D0%B8%D0%BC&f=1003&t=1&v1=&f=4&t=2&v2=&x=28&y=12&f=21&t=3&v3=&f=1016&t=3&v4=&f=1016&t=3&v5=&bf=4&b=&d=0&ys=&ye=&lng=&ft=&mt=&doi=&dt=&vol=&pt=&iss=&ps=&pe=&tr=Cyr-Common&tro=&cc=b3&i=1&v=tagged&s=2&ss=0&st=0&i18n=ru&psz=20&bs=20). It's far from being complete, but it's the largest online source I know of. 

I don't think there are any indexed/searchable sources at all.
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> .