Hi Rmatlin,
We understand you have questions about your message settings. We're happy to help.
The options you're talking about are not present on iPad. You'll only find those options on an iPhone. MMS messaging is a service from a wireless carrier for sending media through regular SMS text messages. The Group Messages option combines messages that are sent through the regular SMS message service so they all appear in one thread like iMessage group messages do. Since an iPad cannot natively use SMS/MMS through the cellular network, it does not contain the "MMS Messaging" and "Group Messaging" options in the Settings menu. This is expected.
Now if you're just using iMessage on the iPad through your Apple ID and don't have an iPhone linked to it, we'd recommend trying to turn iMessage off and back on to see if that helps. If you are having the messages forwarded from an iPhone, we'd recommend checking Settings > Messages > Text Message Forwarding on the iPhone to make sure the iPad is still selected as a device to forward to. If it is already on, you can also try toggling it off/back on to reset the forwarding.
If you just have the issue with one specific group message thread and all others are working fine, you can try deleting the thread from your message history so a new thread is started the next time someone in that group sends a message.
Set up Messages on iPad
If you continue to have issues with group message threads, we'd recommend contacting Apple for assistance: Get Support
We hope this helps. Take care!