Regarding $\alpha_S$, the provisional PDF4LHC recommendation is to use
0.118 as a common value both at NLO and NNLO (I favour a higher value of 
0.120 at NLO since most NLO extractions give a higher value and most NNLO 
corrections are positive, but this is not the standard view). The PDFs are 
standardly made available at values of \alpha_S with spacings of 0.001 and with
round values, so are available from all groups at 0.117, 0.118, 0.119 for 
example. We feel it is unnecessary to go to the 4th decimal place and use e.g.
0.1185 since this precise value is likely to change slightly quite soon. 
Also, I feel that PDF fits do have so non-negligible sensitivity to \alpha_S, 
and the MMHT NNLO number for \alpha_S when the world average is included in 
the fit as a data point is 0.1178, very close to 0.118. The other PDF fitter 
would not support this as strongly, by the NNPDF results on \alpha_s are 
extremely close to ours. We are (probably) recommending 0.001 as the 
uncertainty, so 0.1185 is well within the uncertainty band. There is no real 
special reason for 0.001 as opposed to 0.0012, other than it is logistically 
simple given the values of \alpha_S at which PDFs are available. It was also
the value suggested by Gavin Salam when I had discussions with him following
by talk on \alpha_s at the PDF4LHC meeting https://indico.cern.ch/event/343303/other-view?view=standard. 

Regarding quark masses:

So long as it is pretty near the world average then any value for m_t^{pole} 
near the world average is consistent with PDFs (we use 172.5 GeV, the PYTHIA 
value and NNPDF 173.09 GeV). The inclusive top cross 
section does go into the global fits and the fit quality (and best \alpha_S)
does depend on the value, but the small amount of data used so far means the
PDFs extracted do not vary by significant amounts within sensible m_t limits. 

For bottom quarks each group extracts the PDFs using a particular value of 
m_b^{pole}. For MMHT and CT14 this is 4.75 GeV, which is pretty much consistent
with the value suggested up to the renormalon ambiguity, i.e. I get 
m_b^{pole} \approx 4.9 GeV converting from (11), but with an uncertainty of 
maybe 0.15 GeV. However, NNPDF3.0 uses 4.18 GeV as a pole mass. The extraction 
of the PDFs is actually pretty insensitive to m_b, since very little in the fit 
cares about this (see e.g. Table 4 in arXiv:1007.2624). The exception 
is the bottom distribution itself (see Fig. 6 in the same paper). Hence, 
care will need to be taken when using PDFs when the b PDF is important. If 
working in the 4-flavour scheme I don't think there will be any significant 
m_b PDF dependence.   

For charm quarks again each group uses a particular value of m_c^{pole}. 
In this case the MMHT(MSTW) default is 1.4GeV, which again I regard as 
essentially consistent with (12), where I get \approx 1.5 \pm 0.15 GeV. This 
time the lack of perturbative convergence gives a rather larger relative error,
of course. NNPDF3.0 and CT14 use lower values of 1.3 GeV and 1.275 GeV 
respectively, which seem a bit to low for consistency to me, but do 
admittedly give a better global fit. In this case the PDFs extracted are a 
bit more sensitive, with potentially a percent or so variation in standard 
cross sections, though details depend on \alpha_S correlations (see e.g. 
Table 3 and Fig. 7 in the same article as before). The charm PDF at high 
scales is relatively a lot less sensitive than the bottom PDF to mass
variations due to the increased evolution length. So for variations in charm 
mass consistency would really require using the same charm mass in all parts 
of the calculation, otherwise correlations would be lost. This is much more 
the case for charm than bottom. The method proposed in 4.4 would be 
relatively conservative. 

As it stands each of NNPDF, MSTW and CT have previously made sets available in 
different flavour number schemes and will with updated sets (we will make MMHT 
available soon). MSTW made available a variety of sets with different charm 
and bottom masses, as did NNPDF2.1 (with less variation for m_c). 
Again MMHT will do this, and the fits for different pole masses with variation 
of 0.05GeV in m_c^{pole} and of 0.25 in m_b^{pole} have been done and been 
reported. I'm not sure about other groups, though would be surprised if NNPDF 
don't do something similar again. 

Ideally groups should settle on agreed values of masses, and probably on 
agreed msbar masses. However, this is not the current situation. 

I agree with Michael about using the conversion of masses according to the 
order of the calculation.