The analogue meter is actually a milliamp meter, which displays the number of milliamps flowing through the coil in the meter. The different "ranges" are selecting different circuits, to "condition" what you're measuring into milliamps, so you can read it from the scale. This is why you have different scales on the meter display - all the different measuring ranges apply different factors.
When you measure resistance, your meter is trying to make current flow through the external circuit - which is why zero on the ohms scales is at the opposite end, because your meter is actually injecting maximum possible current.
Zeroing your meter for resistance readings is compensating for the resistance of the leads and the connections. Since it should all be significantly less than 1 ohm it shouldn't be significant for higher resistance readings, but you can still go through the same process - as Gary said.
Zero on voltage and current readings should be zero regardless - if there is no energy in the circuit, there's nothing to measure. I'm not sure how your zero ohms adjustment is affecting your DC readings - so I guess I've just typed a lot of stuff to say I can't help you.
I love digital multimeters - just read the number, and try not to think about it.
Scott