Hood sensor latch faulty

You don't need to remove the switch to get rid of the warning, you just have to disconnect it. On some models A4 the detector is part of the latch mechanism, on others (depending on model year) it's on a separate push-in catch near the latch. You can see a cable going to it with a modular connector. Simply pull the connector apart and secure the ends so they can't get sucked into anything. I don't think you need to jump across the wires to get the light to stop, so try it without wiring them together first.

If it's a separate switch from the latch it may be that the switch has just come out, or maybe the plastic bracket that retains it has broken. You may be able to just pop it back in.

I don't think that the wipers are in any way connected to the hood sensor, on my A4 I ran the wipers with the hood open several times on all settings. I think you have a separate issue there, possibly the selector mechanism, although that wouldn't explain the in motion thing. Maybe it's a relay, I'd check the simple things first.

I had a similar issue with a Ford Focus. I know - different brand, but bear with me.

The issue was that the bonnet (Hood in US English) catch was working fine, but that the switch that detected if it was on or off was poorly waterproofed and shorting. Rather than replace the bonnet catch and sensor (a bit of a job, which i've been told is a nightmare in that particular focus model because the bonnet release is really hard to seat accurately) I simply bridged the switch.

I took a 2 inch piece of electricians wire, removed the plug going into the unit from the electrics, inserted one end into each terminal, and electrical-taped the crap out of it for waterproofing. Several months on it still reads as "shut". I've checked it a few times, to ensure that it's not charring or burning, but it seems fine so far so am planning on leaving it as a longer-term fix. I plan on running this car into the ground, so I'm unconcerned about resale.

Now, dependant on your model of vehicle this may or may not work for you, but if the switch setup is the same (I.e. simple current flow = shut, no current = open). I also counsel caution because I don't know how this would affect the electrics, but it's worked for me on a 2005 focus so far.

I agree with others in this, in that it sounds odd that the wipers are linked into the bonnet catch. This might be a good way to test that out.

I had the same problem on my A3. Hood sensor was flashing on the dash. Couldn't set the alarm. Wipers wouldn't work unless moving. I tried cleaning the sensor but it didn't work. So I cut the wires going into the sensor and capped them with electrical tape. Didn't cross them. Now the hood sensor is off. Wipers work. Alarm sets.