Actually, I think you already answered most of the questions yourself...
As you state, the .geodatabase files are ESRI's new Mobile Geodatabases based on SQLite that ESRI introduced at 2.7.
I have seen similarly uniquely numbered files in TEMP in older releases, and these are likely related to some form of local caches, either for imagery or vector (tile) based services you are using in one of your projects, or alternatively, as you stated, the ArcGIS Indexing Service may be involved in the creation of these, although I doubt it would create this much of crud.
I guess ESRI substituted some other file based caches for the new format.
Of course, if these caches do eat up your entire drive, I think it is time to report this to ESRI as a potential bug. That shouldn't happen.
Anything in TEMP is by definition "scratch", and can be safely removed. If not, it is a bug in the software. Do note though, that some of these files may still be in use and locked, e.g. by the Indexing Service or whatever is using them, and thus not removable.