coexpression = correlation and with wgcna you can use different correlation measures to build your network, i.e. pearson, spearman, bicor, mutual information, etc...

1-Does it mean that, for example, D930050H05RIK is coexpressed with GCLM and so is FMO2? But FMO2 is not as coexpressed with D930050H05RIK?

basically yes. Although all genes in a module share some degree of co-expression (they are all connected), to improve the readability of the plot and highlight the hub genes (GCLM and ZFP307), you show only the edges having the highest correlation values.

2-If I had to validate one of these biomarkers, should I choose GCLM, considering it has a higher centrality than these other two genes?

Yes, you should focus on the hub gene GCLM

3-another output from WGCNA are the GZ.trait values, which from what I understand is proportional to the correlation of the gene with the trait of interest. Should I choose the biomarker based on this, or according to question 2, or both?

This is a tricky question because hub gene do not always have the highest correlation with the traits of interest, especially if the traits are continuous variables