What do you mean by "pool"? A meta-analysis can be run on studies reporting a Pearson correlation. Initially, the correlation is converted into a Fisher's z-scale. Depending on which meta-analysis software you are using, it might be you or the software that does the conversion calculation. See https://www.meta-analysis.com/downloads/Meta-analysis%20Effect%20sizes%20based%20on%20correlations.pdf.
Yes I believe you do include non significant (the demarcation between sig and non sig is arbitrary anyway) as well as significant. Because significance can be an outcome of small sample size - you could have ten studies with non-significant correlations but all with the same sign, and combining them would show an overall consistent and significant effect.
Absolutely agree with John Dawes. In the preliminary analysis one should consider all eligible studies for pooled correlation coefficients. Later a sensitivity approach may be adopted after excluding studies with very low sample sizes and high risk of biases, if essential.