tl;dr: How much did you pay for that shirt? How often do you wash it? Why would you bother? Shirts just get dirty anyway. Can't you get another shirt?
Are there any benefits of washing the outside of my car?
Do you live in an area without salt, birds, tree sap, pollution, abrasive sand or road tar? Do you also drive around with brake pads that don't emit corrosive dust? If so, then you might be okay to avoid washing. If not, then yes, there are plenty of benefits.
Do I really need to wash the exterior of my car every week?
Does it damage my car if I don't wash it for a month or two?
What's practical for your situation? If you can't wash the car for a while because of life circumstances, the point is moot.
Here in the Northeast USA, car washing is actually seriously important. For example, salt is a year-round fact of life here in Rhode Island. In the winter, it's used to treat the roads during snowy weather. In the summer, it's fun to go to the coast (admittedly, Rhode Island is basically all coast but still...) where the wind is picking up all that salt spray and misting it all over the car. Even better, seagulls drop blobs of acidic smears of grossness all over that deliciously salted car.
Some people will try to argue that clearcoat and paint will protect the metal of the car. While that is technically true for a while, neither is sorcery. Any sort of corrosive or abrasive will eventually wear through that protection and start rusting. As a practical example, that gravel shot into the hood of your car by the truck in front of you may have chipped the hood down to the metal. It almost certainly took off the clear coat and a layer of paint.
Any sort of washing will get all that nastiness away from the car with the side benefit of making it look nicer. And remember, anything is better than nothing: I spray out the underside of my car while I'm washing it. I'm not using special undercarriage spray or anything like that, just a straight up garden hose. The startling amount of sand and salt that pours out from under the car could be used to make little castles in my driveway.
It's also a convenient time to just walk around the vehicle and check for any issues. Is that piece of trim coming loose? How long has that screw been missing? Here's a rock chip: I better get some touch-up paint into that right away.
All of the above is bounded by reality. In the winters here, sometimes it's just too cold to wash the car for months at a time. Some of those times, washing the car is effectively moot anyway since all that dirt is getting hidden under layers of snow....
Pro tip: a clean car is much more pleasant to work on. Even if you don't handle your own maintenance, the techs who do will be grateful that they aren't working in a gross muddy rust pit.