Time code generated in file-based recorders is fundamentally different than time code on a tape recorder. Whereas a tape recorder has a continuous time code reference for each location on the tape, a (broadcast wave) audio file only has a single time code value—from which all time code values are calculated.
"Phantom powering" is a method of providing power to microphones by applying a voltage to the same wires that carry the audio signals. Phantom power can be generated from mixing consoles, mic preamplifiers, or in-line phantom power supplies. All of Sound Devices microphone inputs supply phantom power.
Rechargeable AA batteries for internal powering of pro audio gear have historically been poor performers. Alkaline primary cells, and more recently Li-ion cells, are usually the preferred cell chemistry. Fortunately, battery chemistries have evolved -- in some applications there is actually a runtime advantage when using the newer NiMH rechargeable batteries with portable gear versus alkaline cells.