Yes, you will be able to transfer your contacts without having to manually add them. Google will automatically sync them when you log into your Google Account during setup. You should be able to view all your contacts here: https://contacts.google.com/preview/all
Yes, you will be able to transfer your photos. I would recommend backing them up to Google Photos via the Google Photos app (if you have not done so already). You will be able to re-download them from Google Photos later. If you currently use Google Photos, you should be able to view them here: Google Photos - All your photos organized and easy to find.
Yes! You will absolutely be able to transfer your number to the new G4. There will be an option to move your old number to this phone. Your old phone will then be deactivated. See this document for more information and a super-helpful guide: Activate My Phone .
I have not personally upgraded from a G1 to a G4. However, I am positive the G4 will be much, much faster! Also, regarding dropped calls, the G4 should perform better. In the meantime, could you explain exactly when the phone is dropping? Is it on Cell or WiFi?
I hope that helps!