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If you can fly an NDB/VOR/ILS approach without a map then you can fly a GPS without one too, but obviously if you are flying a GPS approach then you have a moving map in your cockpit!
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Not necessarily, there are GPSs out there that do not have moving maps and yet are approach certified.
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If your GPS screen fails then you obviously would not be able to continue the approach anyway since there would be no way of telling if the approach is armed, if you have RAIM and no way to identify the fixes.
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Not necessarily, there are many installations that use an external MFD that displays everything you need. As long as you got the approach loaded and armed before the GPS screen died, your home free.
Even in installations that don't have an MFD to display the loss of RAIM, you don't always need the GPS screen. When we're flying a GPS approach, we can have the fixes displayed on the MFD, the distance ticks down on our HSI, and if the FMS loses RAIM, then the HSI (which was displaying course information in white to signify GPS data) reverts to green needles showing that it's using VOR data. We never need to look at the FMS screen.
The problem with GPSs is that there is no standardization. Every installation is different. In some installations, you have to have the GPS on a specific page to get all the required information. In other installations, you could have a movie playing on the GPS screen for all the good it's doing you. There is no general rule that you can use for what needs to be shown on the GPS screen.
Do you need to see distance to the next fix? Maybe maybe not. Some GPSs show each step down as a seperate fix in the database. Distance is trivial in this case, since the GPS will indicate when a fix is reached, at which point you step down to the next one. Other GPSs don't show each stepdown as a seperate fix, and require you to monitor distance from another fix to determine when you've reached a stepdown.
Can you have the GPS in map mode on an approach? Sure you can. There is no rule saying you can't. To have such a rule would be dumb, on some GPS models, the only way you can get the required data is to be in map mode.
As for the accuracy of following a magenta line on a moving map as compared to an HSI, try zooming all the way in on the map of a 430/530. It's more precise than an HSI is, but the refresh rate isn't very good.