I wonder if it's ok for me to add new player's emails to my mailing list. Should I display a notice or something before doing so? I don't see these a lot in popular mobile games, but they send emails anyway from time to time.
You should ask before signing users up for a email list. Doing so doesn't harm the users who are interested, but can make the users who aren't interested upset with you. So it's generally the "nice" thing to do.
In some jurisdictions, there may be law that require you to only offer opt-in email lists, or require you to present opt-out notices in certain ways, et cetera. The most recent such law I'm aware of is the EU's GDPR, which has some sections pertaining to email marketing.
You should consult your lawyer for specific legal advice regarding your situation and the game you plan to ship.