Redshift Versus Luminosity I understand that there is a relation between the proper distance of a cosmic object and its "measurable" redshift, i.e. once you know the value of the redshift parameter z, then you actually know how far this object from you. If this is correct then why do have to find another way of measuring the distances and use the luminosity "method"?
 A: Cosmic expansion is just one source of red shift, however, cosmic expansion dominates for all but "nearby" objects.
So can you use red shift alone for remote objects? Yes, you can.
However, the distance scale must also be calibrated, for the Hubble parameter must be experimentally determined. Hence the interest in "standard candles". 
See also https://ned.ipac.caltech.edu/help/zdef.html
A: Since we are dealing with very large values with very large errors, yes one can estimate how distant objects are from only the red shift.  At least you will have a ball-park number, two methods to try are
t = 2(Universe age Gyrs)/(1+(1+z)^2)
which will give you the time of the emission from the big bang according to Carmeli. So if we use a universe of 15 Gyrs and a redshift of 0 we calculate the age of the emission from the BB of 15 Gyrs. 
The second method requires more assumptions where
t = (2/3) x (1/H_0 x Omega_m x (1+z)3/2)
Here H_0 is the current Hubble constant, Omega_m is the current, normalized matter density, z is your redshift and x mean multiply. This is from the P.J.E.Peebles book, page 102. You can select a H_0 of anywhere from 62.3 to about 73 and an Omega_m of anywhere from 0.02 to 0.3. Unfortunately, such freedom gives you most any distance you like.
Neither approximation is really much good beyond a z=2.
