  Correlations between the QCD coupling \alpha_s, the gluon condensate <
\alpha_s G^2 >, and the c,b-quark running masses m_c,b in the MS-scheme are
explicitly studied (for the first time) from the (axial-)vector and
(pseudo)scalar charmonium and bottomium ratios of Laplace sum rules (LSR)
evaluated at the \mu-subtraction stability point where PT @N2LO, N3LO and <
\alpha_s G^2> @NLO corrections are included. Our results clarify the (apparent)
discrepancies between different estimates of < \alpha_s G^2> from J/\psi sum
rule but also shows the sensitivity of the sum rules on the choice of the
\mu-subtraction scale which does not permit a high-precision estimate of m_c,b.
We obtain from the (axial-)vector [resp. (pseudo)scalar] channels <\alpha_s
G^2>=(8.5+- 3.0)> [resp. (6.34+-.39)] 10^-2 GeV^4, m_c(m_c)= 1256(30) [resp.
1266(16)] MeV and m_b(m_b)=4192(15) MeV. Combined with our recent
determinations from vector channel, one obtains the average: m_c(m_c)= 1263(14)
MeV and m_b(m_b) 4184(11) MeV. Adding our value of the gluon condensate with
different previous estimates, we obtain the new sum rule average: <\alpha_s
G^2>=(6.35+- 0.35) 10^-2 GeV^4. The mass-splittings M_\chi_0c(0b)-M_\eta_c(b)
give @N2LO: \alpha_s(M_Z)=0.1183(19)(3) in good agreement with the world
average (see more detailed discussions in the section: addendum). .
