Apart from the mandatory emergency gear that should be in cars (early warning devices, fire extinguishers, visibility vest, first aid kit) what other tools should I get?
Should I get a tire pressure gauge or something? Are there tools which could help me diagnose problems?
Aside from what you mentioned, I always carry:
Portable air compressor (12v)
Small tool set (common wrenches, screwdriver, grips, electrical tape and zip ties).
A good flashlight
Spare cellphone charger (both DC and AC if car does not have USB plug)
List with emergency contacts in glove box.
Rope in case I need to be removed from a ditch or towed (happens more than I'd like to admit).
Copy of keys to open trunk and doors on wallet.
Money in trunk inside plastic bag ($100 is good).
Jumper cables that are are at least 3 meters long.
Towel
Spare change of clothes (includes shoes and socks)
A bat or something to defend yourself in case of zombie apocalypse (or anyone tries to hurt you and you can't run).
I'm a bit obsessive about this, but these are things I've needed over the years (including the bat).
To be a little more pragmatic than the other suggestions... what classifies as an emergency can be subjective.
To me the emergencies I have experienced in a motor vehicle typically come in the form of flat/blown tire or running out of gas. Because of those experiences I keep the following in my car at all times:
Tire Changes
Out of Fuel/Dead Car
Poncho and or a coat - Anyone who has had to hoof it a few miles down the road to a gas station will recognize the wisdom in always having something in your car for adverse weather just in case you find yourself forced to brave the elements.
Emergency Blanket - I've fortunately never had to use mine but I've always kept a survival blanket in my glovebox on the off chance I ever have to spend an impromptu night in my car when its cold outside. These blankets are cheap, very small but could save your life in the right situation. I've also known (male) friends to pack Panty Hose in their trunk for the same purpose as hose helps to trap body heat.
A friend gave a really good advice, since most smartphones are quite fragile these days (just look at the number of shattered displays), it is a risk that they break if you are in an accident. He always keeps a "dumbphone" that has a long battery life in the glove compartment or in his motorbike (think Nokia 3310, basically indestructible).
This way if you do break your smartphone and need to call for ambulance you have a spare one (no need for a sim-card in most countries, so no extra cost there).
I usually carry something I can use to break the windows in case of emergency. This can be a life hammer. I also have either a small knife or a dedicated tool to cut the seatbelt if needed. Both window breaker and knife I place strategically in the car so that I can access them quickly in most situation, including when the car is flipped upside down. Finally, I always have an emergency/space blanket in the car.
I would add
My equipment:
Hardware:
Spares
Clothes / hygiene
Others
Seasonal stuff
Where I live - snow chains can be helpful. Check local rules.
If you can't have chains, a piece of carpet can help you give your tires some grip when your wheels slip. Or (non-clumping) cat litter.
In winter - some "emergency" food (Clif bars etc), and a sleeping bag.
Stuff I'd recommend carrying (some of which is required in certain jurisdictions, other things are optional everywhere)
In the car:
In the boot/trunk/load-space:
It depends on usage but in rural or forested/swamped areas — a winch (kind of manual) is probably a good idea.
Also I'd like to install amber flashing lights activated by shock sensor/gyroscope (in case of flip over) but not sure if it's legal.
This is something I learned from my uncle. Every once in a while you buy stuffs on boxes. Vehicles aren't that plain-ish compared to boxes.
