
I think there are a lot of variables that need to be considered with your question, but if you are working in DevOps or agile, testing would be ongoing. As far as who is responsible, it's important the entire team is dedicated to quality. As far as regression testing, there is probably a lot of automated testing you can introduce to reduce the manual testing work load.
My experience with testing is minimal, but there are a lot of great practioners and experts on our TechWell Hub Slack workspace. I'm sure if you drop this question in one of the channels there, you'll get feedback from people a lot more experienced than me! You can join at hub.techwell.com.