Typical SPL, don't really know numbers, the main thing is to get a good SNR so the background noise in the space enters into the equation.
Omni speaker, I see you are posting from .nz. There is such a speaker at the Acoustics Research Centre at the university in Auckland. I was there when it was purchased.
I think you can get pretty good RT measurements with a non-omni speaker, I would use one on its back on the floor. The other parameters you mention should be measured with an omni speaker, as they are early-field parameters. And no, don't measure with the source in a corner, unless that is where your natural source will be. These parameters will vary strongly with position.