Have you tried measuring the voltage at the actual PICAXE power pins when everything connected?
Well worth a double check with a multimeter. A resistive (e.g. dirty or loose) connection could affect things.
If I were doing this I would tediously be checking the quality of continuity for ALL the wiring with a multimeter.
Sometimes something really basic can give you a "Doh!" moment.
Re: Capacitor for the GP2D12
There are several variants of this device - and several varieties of Data Sheet.
Some mention it, some don't.
I don't know how important the capacitor is, but as it gets a mention then it can't hurt to include it.
http://www.farnell.com/datasheets/62635.pdf
Page 8 Para 7-8
http://www.farnell.com/datasheets/62636.pdf
Page 3 para 7-8
Two other Data Sheets I looked at didn't mention it.
Capacitor goes across the GP2D12 power pins, and it implies you puit this capacitor as close as practically possible to the GP2D12.
I'm not suggesting this will cure the problem, just that it may need it for best performance once you have it up and running.
I also see from the 4 varieties that i have looked at that Sharp recommend 4.5 to 5.5V operation.
I think 3 AAs will be too feeble for prolonged oepration, though should be OK for brief testing - assuming you batteries are OK.
I would certainly disconnect the other bits 'n' pieces while you get this bit sorted.
I've seen numerous examples people going head-first into a monster breadboard and then wrestling with a problem. Always build up in stages - even the Big Boys do that.
An option at a later stage would be to have the PICAXE being able to power-up the Sharp so you can use it when you wnat to and not waste power. I'll leave that for others to tell you how.