Electrical – this thing that looks like a light socket but isn’t

The socket itself is probably shot, but you should be able to fix the range hood by replacing the socket. It looks to be held into the chassis by a couple of retaining clips and should come out pretty easily from the back, although you'll probably have to remove the cover to take it out properly. It looks like this part is what you would need to fix it (but don't take my word for it -- make sure to do your own research and verify that the replacement part looks the same as the original). I found this by Googling for "Nutone Allure range hood replacement parts". If it is indeed the correct part, then it should just plug right in.

In general, NO. Replacing a fuse with a higher-amp fuse is a very bad idea and can lead to fires.

Fuses (and circuit breakers) are rated such that they blow or trip before any part of the circuit gets to a dangerous current. You rate the fuse or breaker based on the lowest current rating of all the devices and wires in the circuit .

In your case, the manufacturer of the connection unit determined (somehow) that the "weakest link" in that circuit can only draw 5A before it exceeds its limits. Putting in a 13A fuse would thus allow that weakest link to draw almost 3 times what it may be designed to, which leads to increased heat and then fires.

There is a reason your fuse is tripping and you need to figure out what that reason is. Perhaps there is a problem with your lamp post, the bulb, one of the sockets, or the van charger. Fix the underlying problem, and you won't have any more blown fuses.