First let me start by saying I know this question has been answered with various nuances, but when it comes to safety I want to make sure I have the most appropriate information for my situation.
We're in the process of moving into an old house (renting, currently on a 6 month lease). When I did the walkthrough I didn't notice that every outlet was a 2-prong; it just wasn't something I'd even considered. I have a number of 3 prong devices and I want to make sure I'm using them safely (both for me and the device).
The only place that there is a 3 prong is in the bathroom. It is not a GFCI (none in the whole house). However I tested it and with the neutral and hot connections I get 120Vrms. With the hot and ground I get ~10-12Vrms and with neutral and ground I get ~1Vrms. I take it to mean that it is not actually grounded and a previous tenant replaced it so they could plug in a 3-prong device...
Also very concerning is the the fridge is plugged into one of those crappy 2->3 adapters and I think the ground pin is just floating and not actually connected to anything... this seems wrong at best and dangerous at worst.
I looked into adapters but the screws at least on the outlets' faceplates are not grounded. The house was built in 1921 and I have no idea how much wiring has been done since but I think it's safe to assume there probably isn't any grounding wiring. I am, however, able to get a clean 120Vrms signal from some of the pipes and various metal around the house, FWIW.
We've asked the landlord and they want nothing to do with replacing any of it and suggested we "buy some cheater plugs". Which, yes, will technically work but it doesn't feel like a good solution. If I were an owner I'd probably just pay to have it all redone. I'm in Seattle, if that affects any of the answers.
What should be done about the fridge? I was thinking a minimum of adding a GFCI for safety. I do worry that the fridge will trip it, which I've heard of happening.
What can be done for the other outlets to make them safer for me and my devices?
How much of a problem is this, really? Is it something that we should be concerned about to the point of moving in August? We were hoping to live here for a few years at least, but it isn't likely to be properly rewired any time soon.