I have a flashlight, jumper cables, reflective warning triangles and a tire gauge right now. I’m thinking of buying an air pump because I won’t always be near a gas station if my tires get flat. But do I really need a fire extinguisher?
I want to make sure I have everything I need, but I don’t want to go overboard with stuff I’ll never use.
If it makes you feel better then buy one. As for air pump why not just get the cheapest AAA membership?
That’s what I did. Your other thread says you had a tire 10 psi low. There’s a good chance you have a slow leak on that one. Check that one more often.
I’ve never needed a fire extinguisher. Even back in the '60s when my buddy’s Studebaker engine caught fire while we were sitting talking to some girls (that’s how we met girls in those days). We just smothered it, started the engine again, and kept looking.
But if it makes you feel better, get a tri-rated extinguisher. Putting the wrong type of suppressant, like water, on a gas fire will turn a small fire into a charcoal car. Gas floats on water, and the fire will simply spread.
In North Dakota, before cell phones, we always had severe weather gear in the trunk. Go off a highway in the winter and you could be there in subzero weather for days before someone stumbled on you. People froze to death every winter.
Truly, the only things you need these days is a cell phone and an auto club membership.
Don’t you have a spare tire? A can of fix a flat is probably better than a tire pump.
On long trips I carry the following:
It all fits into a nylon travel bag in the trunk.
Of course my cell phone and AAA membership are my best friends.
I;d add a window cracker, such as This one from Amazon. And keep it near the driver’s location, with a flashlight.
However, I’d been driving for 40 years before I bought one…
Some guys still do that today, but now it’s not called “meeting girls.” It’s called “stalking.” 
CSA
When a problem arises, no matter what you’ve selected to keep in your car, You’ll need something that you don’t carry.
My daughter carries a pink keychain mace dispenser! I bought it!
You could also carry some distilled water, some kind of long lasting sealed food, and a sleeping bag or articles of warm clothing.
CSA
Everyone has lots of ideas, so I’ll add just one. A handful of disposable nitrile gloves. You can buy them lots of places, sometimes they are called exam gloves. They are like the ones your doctor uses, except you don’t need yours to be sterilized. Hardware stores, drug stores, maybe even dollar stores have them.
There are all sorts of things that come up with operating a car, and some of them are really dirty or wet or oily, and when you are wearing those gloves it’s a lot easier to just take care of business without getting your hands filthy.
And, if you find them 2 years from now, stored away, and you decide you’re never going to use them in the car, they are great for cleaning the bathroom or the kitty litter box.
I have a Dahon Helios aluminum 8-speed folding bicycle. Rather than store it at home I just keep it in my trunk. I drive larger GM cars and have plenty of room, and besides I like to explore when I travel.
If push comes to shove I can travel at 18 mph (on level ground) and cover distances fairly fast and with little effort. 
CSA
I call that a four way lug wrench. Not sure of real name.
I go that one better. I took a long cross wrench, cut the sockets off of the two ends 90 degrees from the one I need, and slid pieces of conduit over the two cut-off shanks. They slide in for storage and slide out for considerable added leverage when I need them. It works great. My bad back makes using even a long-shank cross wrench painful, but this does the trick.
NOTE: I’ve had enough experience with conduit to know that if it’s overstressed it’ll progressively bend rather than fail suddenly, so I’m confident that it’s safe.
I also have the usual stuff; a click-stop torque wrench with an impact socket, a breaker bar, a hydraulic jack, leather work gloves, a code reader, a $1 mirror to find the plug under the dash (
), and a plastic garbage bag to put the snow-covered muddy flat tire in. I’m thinking of replacing the torque wrench with a torque-stick, just for space-saving.
For the record, I don’t recommend fix-a-flat. That stuff makes a mess, and a lot of shops won’t work on a wheel that’s been reinflated with that stuff. Although I realize that many performance cars now come with only that and no spare. Blah! 
What you already carry is pretty good, better than most drivers. I’ve never carried an air pump or tire gauge. I just check the tires once in a while and top them off when I’m filling up the gas tank at the local station. I do carry a small-ish fire extinguisher. Other things which I carry
Other ideas to be a pro-active car owner:
I don’t personally consider a fire extinguisher a necessity. I figure my car is insured, so if it were to catch fire, I’d only be concerned about my life (and the lives of passengers).
If you’re planning to change your own flat tires, you might purchase the appropriate tools and a good small floor jack (because the jack and tools that come with most cars are terrible), but if you have other plans, like an auto club membership or someone you can call locally, those tools would be redundant.
If you buy a can of Fix-A-Flat (or one of its competitors), make sure you get a non-flammable formula.
Yup, that stuff is disgusting
I recently replaced tires on one of our fleet vehicles
Turns out one of them had gotten a fix a flat treatment at some time. It wasn’t obvious, until I broke the bead.
I had to load the rim on our golf cart, for shop use, and head over to the wash rack area. There I thoroughly hosed it off. I also replaced the valve stem. So that fix a flat added several minutes to what should have been a very routine job
I think you get the idea but I’d add a first aid kit. You can buy one but then go through it and add antiseptic, some large telfa pads and tape for big wounds, insect spray, etc. Car accidents do happen and sometimes need to stop the bleeding quick. The thing with a fire extinguisher is that a small one is of little use because it gets used up so fast. Even a large one can go empty in a hurry.
Plus 2 on the weather gear for cold, rain, snow, etc. Actually I always have a couple bungee cords too along with a quart of oil. If you do need to change tires and have a trunk full of junk, the bungee will keep the trunk down if you have to throw the flat tire in there.
Cell phone charger and a bottle of water. Maybe a single granola bar of some sort. I have a medical kit in my car with safety glasses, surgical gloves and a mask among the other things Bing suggested. If I stop to assist an injured person I want to able to protect myself from infection. I don’t carry a fire extinguisher, but I like your idea of a compact air pump.
Best thing IMHO, is a roadside assistance plan such as AAA. I got mine after having to change a tire on an interstate in 95 degree heat, never again!
They cover lockout service, jump a battery and other services.
Now beyond that, if in a cold climate, extra clothing-coats, sweaters, blankets.
Desert, extra water.
Fire extinguisher is a good idea.
A flashlight.
And of course always have your phone.