Hi @nyprosell. Thanks for the questions. I will give you my opinion.
You basically just start doing it! Get all you co-workers/teams signed up for Asana and begin using it for all project/task or general communication. You can read a bit more on this here: https://asana.com/guide/team/onboard/move-communications-to-asana It really is dependent on your business structure and what you need to accomplish on a daily basis in regards to internal communication.
From my experience, Slack is great for internal communication (DMs, announcements, brainstorming, …). But anything which requires an action/task/project - should be put into Asana. All communication related to these projects/tasks should then take place in Asana. https://slack.com/apps/AA16LBCH2-asana
Yes, but you will need to go through a 3rd party app. Mailbutler is one that is free (with restricted options) and it works quite well with Apple Mail and creation of Asana Tasks within. https://asana.com/apps/mailbutler