You can infer some of the answers to your questions from the BATS exchange's market data page and its associated help page. (I'm pretty sure a page like this exists on each stock exchange's website; BATS just happens to be the one I'm used to looking at.)
Matched Volume section refers to all trades on a given date that took place on "lit" exchanges; that is, where a public protected US stock exchange's
matching engine helped a buyer and a seller find each other. Because there are exactly 11 such exchanges in existence, it's easy to show 100% of the matched volume broken down into 11 rows.
FINRA & TRF Volume section refers to all trades on a given date that took place on "non-lit" exchanges. These types of trades include dark pool volume and any other trade that is not required to take place in public but is required to be reported (the
TRF) to FINRA. There are three venues via which these trades may be reported to FINRA -- NASDAQ's, NYSE's, and FINRA's own ADF. They're all operated under the purview of FINRA, so the fact that they're "located at" NASDAQ or NYSE is a red herring. (For example, from the volume data it's clear that the NASDAQ facility does not only handle NASDAQ-listed (Tape C) securities, nor does the NYSE facility only handle NYSE-listed (Tape A) securities or anything like that.)
The number of institutions reporting to each of the TRFs is large -- many more than the 11 public exchanges -- so the TRF data is not broken down further. (Also I think the whole point of the TRFs is to report in secret.) I don't know enough details to say why the NASDTRF has always handled more reporting volume than the other two facilities. Of course, since we can't see inside the TRF reporting anyway, it's sort of a moot point.