Linux

/var was originally for files of varying sizes, but today it's more of a general purpose storage for the system, such as log files. It used to make sense to have this as its own partition as read and write operations were generally expected to be small but many, as opposed to few and large for the rest of the storage areas. With its own partition it's easier to adjust the filesystem to accomodate the I/O. Today it's mostly used for logs.

/local used to be similar to /usr/local on some systems, but that's not really the case anymore. It's a directory we use at work for local stuff, as opposed to /global which is shared with the entire server cluster.

You can have any directory as its own partition, just make sure the mountpont reflects it. /home is a very common example of this - using this as a mountpoint instead of just a normal directory named /home prevents regular uaers from filling up the root filesystem and borking normal operation.

Swap is what your PC uses when it runs out of RAM. It can be a partition, or it can be individual (large) files. As an example, I have a rather huge and demanding factorio save which takes up more memory than I have on my laptop, so when I want to play it I have to add additional swap space. It's similar to what windowa refers to as the pagefile. It's slow compared to RAM, but it enables the PC to operate relatively normal despite being bogged down with loads of allocated memory.

I don't know the answers to the other questions but yes swap is important, without it as soon as your system exceeds the RAM available to it it will freeze entirely.