I think the "risk of bias assessment" is an essential part of systematic review and Meta-Analysis. There are different tools in JBI website for assessing experimental and quasi-experimental studies.
The risk of bias assessment is essential and should be displayed in your systematic review or meta-analysis. The assessment is conducted to objectively assess the quality of each included study, and a higher risk of bias could infer limitations on the topic itself. Please adhere to the Cochrane systematic review guideline for more detailed explanation.
Compulsory....Because the assessment of RoB could implicate on the findings of your review. You should point out types of bias and potential confounders of the primary studies you selected. You should interpret how the resultant bias could implicate on the findings.
I think so...There are lots of tools for assessing RoB. I found meta-analysis studies synthesizing correlation coefficients do not conduct RoB assessments though. Please see the Cochrane's guideline and the updated version of PRISMA (2020). JBI might be another good source for your study.