How many photons does it take to determine the existence of a distant object? This applies to any object, but I see the recent discovery of the oldest, most distant galaxy and it started me wondering what the limits are. Presumably you can do better with a bigger telescope and longer exposure, but I'm curious about literally how many photons you need to collect to decide whether or not something is actually there. And then presumably you need a bunch more to determine its red-shift and angular size and rotation etc.  
I have absolutely no feel for these numbers.  Is it tens of photons or a millions of photons?
What matters is how many photons you collect versus how many you would expect to see if the object wasn't there.
Photons would be present, without a source, for a variety of astrophysical (e.g. diffuse background) or non-astrophysical (night sky brightness, dark current) reasons, so you need to be able to rule out the null hypothesis that what you have seen is consistent with no object at all. In principle, if you expect no background counts, then the detection of one photon is significant.
The lowest backgrounds tend to occur in space-based X-ray and gamma ray observations, where the detection of a few photons is often taken to be evidence of the presence of an X-ray source.
The backgrounds tend to be higher in optical observations. More photons from the source are needed to get a significant detection, because the variance in the background is proportional to the expected background.
Determining anything beyond detection typically needs an order of magnitude more photons because you are slicing your data into several positional or wavelength bins and you need significant detections in those individually. For rotation you need both position and wavelength, so I would say you need yet another order of magnitude increase to be able to say anything.
It is hard to write anything but a vague, general answer, because the details depend exactly on the telescope, the detector and the type of object being observed.
For example, one way of looking for very distant objects is to look for mere detections of high redshift Lyman alpha emitters. To minimise the background you use a set of narrow-band filters which would select the Lyman alpha line over a narrow range of redshifts. Detection of a source through one filter combined with non-detection in adjacent filters tells you it is an emission line object, and assuming it is Lyman alpha you then also know the redshift.