If the connection only drops out for a few seconds, it will play just fine from the buffer till connection is restored, and the refill the buffer once it's back.
Unless playback comes close to the end of the buffer, you should never see a problem.
When starting new or jumping far ahead, the buffer is empty, so you're basically close to the end of the buffer, which is when you see problems then.
I don't know what the max length of the buffer is, but I would guess it's at least 30 seconds and quite possibly several minutes, and will depend on how the app is coded, your connection speed, and how much memory is available.
You might want to check your DNS settings on your network, it's possible both of your issues are related to that.
You should not use the one provided by your internet provider. as they are notoriously flaky
The ones from google or cloudflare are much more reliable.
Cloudflare is probably the better choice, it also has options for blocking adult content if that's something you need.
To find out what DNS you are using, on the aTV, go to settings->Network. Under DNS look for 8.8.8.8 or 8.8.4.4 (google) or 1.1.1.1 or 1.1.x.x (cloud flare)
if you see something else, you should look into changing your setup
You can change the dns server individually on all your devices, but the easiest and most reliable way is on your router. When done on your router, it will push the setting to all devices that connect to it. There are numerous step by step instructions online.
A DNS failure will not completely stop your internet. If your device has the required IP address stored somewhere a connection can still be established.
It's also possible the movie you're watching is streaming fine since the connection to that server is established and working. And the popup happens because the aTV is trying to reach Apple in the background to check for app updates or or something, but can't because it can't find a DNS sever.
DNS (domain name system) is basically a phone book.
All of the switches and network gear between your device and the device that houses an internet site use IP addresses and not names. When you type "google.com" into the browser, your computer asks a DNS server what google's IP address is. it then uses the IP address it receives to direct the requests to the correct computer on the other side.
The adult content blocking works by removing the entry for certain sites. So if your computer asks "hey what's Phub's number" it won't get a response. but if you know the IP address, you can type that in and still reach the site.