(1) debuglevels 4 & 5 do not seem to print anything about what's returned - this might be a more global issue, though
notice how this key gets "Sending back" entry :

 20647:20141016:014349.161 Requested [net.tcp.service[http]]
 20647:20141016:014349.161 Sending back [1]
 20647:20141016:014349.161 listener #3 [waiting for connection]

but this does not :

 20647:20141016:014350.709 Requested [net.dns.record[,www.zabbix.com]]
 20647:20141016:014350.710 listener #3 [waiting for connection]

net.dns.record shouldn't return ZBX_NOTSUPPORTED for failed queries. It does.

how do those queries fail ?

In this case, any DNS error will do - host not found, no record of requested type, etc. This on a 2.2.6 server and 2.2.6 agent, tested with data types character and text, both fail.

I tested also with explicitly configuring two queries to one host, the other with [,\{HOST.DNS\},A] and the failing one with [,failinghost-notexist.our.domain.tld,A]. The selfcheck works, the latter reports just ZBX_NOTSUPPORTED and thus cannot be parsed or monitored.
Actually, any query which returns a positive result works nicely, so that hard-configured fail-test wasn't necessary. Pity, that we don't care about those OK results, it's the fails we want to catch. Nodata() would cause other problems.

what would you expect that item to return upon failure ?
in any case, that's a different issue - could you please create a new bugreport ? as a minimum, documentation is lacking as https://www.zabbix.com/documentation/2.4/manual/config/items/itemtypes/zabbix_agent does not mention what happens upon failure.

OK, I'll file this as another issue, as it very likely is.

Also get this issue - but seems to be just some domain queries:

zabbix_get -s 127.0.0.1 -k net.dns.record[8.8.8.8,example1.com,MX] returns records with no issue.

zabbix_get -s 127.0.0.1 -k net.dns.record[8.8.8.8,example2.com,MX] returns result ZBX_NOTSUPPORTED: Cannot perform DNS query.