This will be only a partial answer - hope it helps a bit.

First: There is a problem with using correlation coefficients as data. It has to do with the distribution of possible values, which violates the preconditions for standard parametric analysis. The traditional solution is to convert the correlations (r values) to Fisher's z (sometimes called r prime). It's a kind of logarithmic transformation. It's easy to look up the formula and apply it manually or in a spreadsheet, but there also are calculators available on the web. Also, some statistics textbooks include tables. Anyway, you make the conversion, then run your analyses (e.g., calculate the mean value of z across studies), then make the conversion back from z to r.

I believe there is some controversy as to whether this procedure is necessary or even desirable, so you may find authorities (wiser than I!) who say not to bother. What I have noticed is that the procedure rarely makes much of a difference. With low-to-moderate correlations, the conversion has virtually no effect; it's only with very strong correlations that the values chance much (becoming more extreme). In many data sets, after all the fussing, you end up with a mean r barely distinguishable from what you'd have gotten if you just took the mean of the original r values. Still, I'd probably use the procedure if running a formal meta-analysis.

Second: When you say "weighted," I think you mean weighted by sample size? That's relatively easy. If you want to weight by sample size, just multiply each input (z value) by the number of cases in the sample, then, when you have the total for all samples, divide by the total number of cases. That way a study with 200 participants will be weighted twice as heavily as one with 100, for example.

This procedure, too, has its share of controversy. It's certainly appropriate when the studies in question are otherwise identical. But often they aren't. Some may be better controlled than others, for example, or involve different populations. And if you have a study with a very large N, it will basically dominate your results. For instance, imagine this scenario: You are studying creativity. Study #1 has a sample of 80 architects who produce sketches of possible vacation homes set in the mountains. The sketches are evaluated by 3 other architects who work independently of one another. Study #2 has a sample of 800 introductory psychology students who list alternative uses for a brick; the number of uses is the outcome measure. Should study #2 really be weighted 10 times as heavily?

Thank your for your answer. CMA transforms the r value to fisher’s z value and then back to r. However, if I enter five effect sizes per study, the software displays a error message” duplicate row found”. I'm wondering if I could combine the effect sizes into one effect size per outcome for each study to run the analysis.