large s.e.'s for coefficients

Albert,

1) Have you looked at the correlation among your X variables?  

The situation you describe, overall model highly significant but large se's on 
each individual coefficient, often arises with multicollinearity.  Variance 
Inflation Factors (VIF's) are one common diagnostic.   If multicollinearity is 
the issue, there should be an even simpler model that fits almost as well.

2) I haven't checked how glmer computes the quasi-likelihood.  If done 
correctly, that should account for the overdispersion.  If not, it may be that 
the lnL is computed without adjusting for overdispersion but the s.e.s of the 
coefficients include overdispersion.  That will also create the behaviour you 
have encountered.

There is one theoretical issue with the use of a quasi-likelihood ratio test.  
 The quasilikelihoods for both models (full and reduced) should be computed 
using the same estimate of overdispersion, taken from the full model.  The 
usual calculations use different estimates for the two models.  That is not a 
practical problem if the two estimates are similar, as they often are.

Best wishes,
Philip Dixon