Hi Queen of Grapes,
There is not a default setting in Mac Mail that would filter out or stop you from receiving emails you were CC'd on. To clarify, if you check your mail on the web, are the missing emails there? If they're not coming in at all, it's likely an issue with your email provider. On the other hand if they ARE visible on the web but not in Mac Mail, make sure you don't have a rule set up that somehow filters them. Rules are custom created and have a lot of leverage in how you're able to create them. You can check for rules under Mail > Preferences > Rules.
Regarding the second issue, that is much more straightforward. Under Mail > Preferences > General, you should see a prompt for "Enable message follow up suggestions". If that's on, toggle it off.
Finally, if the app is slow or lagging, is it only in Mail or system wide? If you notice sluggishness across the system, boot your Mac up into safe mode and test there as well. Do things run slowly in safe mode, or after you boot into safe mode and then restart again to exit it?
Let us know and we'll continue from there.
Cheers!