With the construction "en/ett sorts..." It's common to take the indefinite article of sort, rather than the second noun, but both ways are perfectly fine.
When it means 'type of'. You can also use it as a filler word much like like or kinda in English. Det är typ omöjligt att tro 'That's like impossible to believe'.
On the other hand sorts has the genitive in it so it's en sorts bok but en typ av bok.
No, not really. I mean, you could say things like
- political system = politiskt system
- societal system = samhällssystem
... and so on, but system by itself is neutral.