Ps4 – Are Dark Souls III PS4 Save Files Compatible Across Editions

Basically; no.

While the story might share similarities through the references of "the old gods" and the "chosen one", with perhaps a few old faces thrown in, the story of DkS1, DkS2, and now DkS3 are not really related.

If you want to follow the Lore, and the history of the series, then by that standard, you may need to play the older games to follow that through, but there are a lot of YouTubers that already have covered a lot of the DkS universe.

EDIT: to slightly contradict my original answer, the community is (unofficially) calling DkS3 the "1.5" of the series. There are similar areas like the Hub and Anor Londo, and a few old faces thrown in, but the primary storyline is very different to the other two games - focussing on the themes of fire, rather than darkness.

PS4 controller acts like a DirectInput controller by default, which is being phased out due to Xinput being superior. While Dark Souls 3 might work with DInput (most games that support controllers also "support" DirectInput) controller, the buttons will be counter-intuitively mapped and it won't be an enjoyable experience.

There is no reason why you shouldn't use DS4Windows with a PS4 controller on PC though, as it emulates an xinput controller while also opening a lot of customization options, like key rebinding including the touchpad, as well as lightbar color change at different battery levels, mouse simulation, etc.