Safety differences between the various types of outlet (two-/three-prong, grounded/ungrounded, with/without GFCI)
1 is against code
2 is allowed provided the correct stickers are applied ("GFCI protected" and "no equipment ground")
3&amp;4 allowed and safe provided it is not a bootleg ground.

Ground and neutral is only allowed to be connected at the main panel. Disconnect all those bootleg grounds from the neutrals in all other places.

You can put the GFCI protection at the panel by using GFCI breakers. The other option is GFCI receptacles and if you do things right you can protect outlets downstream.