Sorry to register for XDA for the sole purpose of resurrecting this old thread, but this thread is the #1 result on "normal" Google search for the search terms below and some extra info/questions may (or may not) be useful to other people doing the same search.
Search terms = Nexus 10 bluetooth
Like the folks who posted here previously I was also looking for the answer to "What Bluetooth characteristics/version does the Nexus 10 support?"
I ran into some info from 2014 in an ArsTechnica article where they talked with Dave Burke, the Google Nexus program manager.
That info, combined with the info above, leaves me unsure which of the following the Nexus 10 supports:
Bluetooth 3
Bluetooth 4 without BLE
Bluetooth 4 with BLE if you make a minor mod to the software environment yourself ("hack")
Bluetooth 4 with BLE supported by the default Android upgrade patches
BLE is the Bluetooth Low Energy feature of Bluetooth 4.
One paragraph from the ArsTechnica article of June 27, 2014 follows and a way to find the original article is below that. The person referred to ("he") is Dave Burke of Google.
"He also enlightened us as to why devices like the Galaxy Nexus, 2012 Nexus 7, and Nexus 10 don't officially support Bluetooth 4.0, even though the hardware itself is capable of it - Bluetooth LE was introduced in Android 4.3 and is crucial to extending battery life in devices like Android Wear smartwatches. Burke explained that supporting new wireless standards in devices often requires those devices to be re-certified by the FCC and other regulatory bodies. In the case of those specific devices, Google decided against recertifying them in the countries that required it, and decided not to support the feature in some countries but not in others."
On arstechnica.com search for "nexus-program-manager-says-nexus-devices-cant-ever-go-away" .
I would suggest that in the last line of the paragraph above the first "not" shouldn't be there and is a simple mistake while typing ... otherwise the sentence does not make sense grammatically.
Anouther (sorta) link to an earlier, related article from around the time that the original posts were made in this thread is ...
On arstechnica.com search for "review-android-4-3-future-proofs-the-platform-with-multitude-of-minor-changes" .
The top article says Bluetooth 4 came in with Android 4.3. It is now 2016 and my Nexus 10 is at Android 5.1.1 with strictly stock upgrades/patches, but I am unsure about the country certification stuff that is mentioned in the article paragraph above. Could Google be releasing slightly different versions of Android to people based on their IP address (country)? That is what the paragraph seems to imply.
Not that it really matters, but my Android version/build/etc. is listed below. I am in the USA and my IP is easily identify and as a USA IP.
Android version: 5.1.1. Android security patch level: 2016-04-01. Kernel version: 3.4.67-g84ad5a4.
I currently do not have any Bluetooth devices, but I am thinking about buying 1 or 2.
As a newly registered XDA member (first post) I cannot post links. That is why the "search for" "links" are shown above.