GPS antenna current is monitored on some units, and with the very few, there's an external IF converter and you actually see a frequency over the DC bias of the antenna connector. Passive GPS antennas almost don't exist to my knowledge - the cable run would need to be so well shielded to let that tiny signal actually make it to the unit it's just not practical. I've seen antennas not specify requiring any sort of power, but every one of them has had an LNA inside that requires a DC bias.
As for the Trimble unit, default baud is usually 9600 or 4800, and there's a chance it uses the TSIP standard to communicate which is not ascii characters but is raw bytes, often with slightly different communication settings (Been working with a Trimble Resolution SMT that uses normal 8 bits one stop, but uses an odd parity bit). Often modules will also have an NMEA 0183 standard output mode, but it may not be the default. I'd try TrimbleStudio or Lady Heather to try and talk to it if you are at least seeing activity on the output - but if it's just the waveform you showed, you're not going to get data from that. edpalmer42's post a couple back may help with getting data out through this line - sounds like a likely board defect.