Please disable battery optimization for the Android app to receive push instantly and restart the notification service (or your device). If this does not help, try to lock the Tutanota app such that it continues to run in the background regardless. This workaround is necessary on some phones, e.g. Xiaomi.
Thanks for your feedback. Actually, we do not log IP addresses. If at all, you can turn on encrypted logging for your own security, but we don't have access to this data. Here's most info: https://tutanota.com/blog/posts/anonymous-email/