Can someone in my house see what I’m doing on my phone thru the Wi-Fi network/settings?

Being connected the same WiFi network will not allow anyone to see what you're browsing on your iPhone. However, sharing an Apple ID or having a compromised Apple ID or some other account compromised could enable that.

If you are sharing your Apple ID then please stop that and have a separate Apple ID

He may be able to see what you were browsing on Google if he had your Google account.