Do my episodes get automatically pushed to Apple Podcasts?
You only need to submit your podcast once to these directories.
Once your feed is submitted and your podcast exists in Apple Podcasts, Google Podcasts, Spotify...etc they will auto discover and fetch your new episodes.
When you make changes or upload a new episode your feed is updated in a few seconds and it contains the new information.
Keep in mind that it's up to directories how often they will check your feed for new episodes and it may take up to 24 hours for them to appear.