It sounds like what you want is a smart phone more than a dedicated GPS/GNSS unit.
1. Some units will, but not all. Battery drain should not be significant but it can vary massively based on the device you buy, for example some GNSS chips will do this on chip (so the rest of the device is not using power) by default and could run for months off a battery. Others devices are highly ineffecent and would drain batteries in a few hours.
2. I've never seen that for use in the bush etc... Google maps etc can do this because the walking tracks/roads etc are all marked. In the bush/mountains thats not true. Straight line is easy, but after that it would need to take into account contours so you don't fall off/climb a cliff, rivers and other barriers, unstable ground, foilage and thick ground cover etc etc etc...
3. Again this depends completely on the device you buy.
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