Steam – report a cheater in Dark Souls 3

Yes, it is against the Steam Subscriber Agreement.

It is against Steam Subscriber Agreement which you agreed by creating a steam account, buying a game or buying/selling something from/on the Steam Community Market.

You may not use Cheats, automation software (bots), mods, hacks, or any other unauthorized third-party software, to modify or automate any Subscription Marketplace process.

Steam Achievement Manager counts as a bot or cheat (TF2 Wiki considers it as hacking), if not just an unauthorized third-party software, therefore is against the Steam Subscriber Agreement.

If you use this software, your account may be terminated:

Valve may cancel your Account or any particular Subscription(s) at any time in the event that (a) Valve ceases providing such Subscriptions to similarly situated Subscribers generally, or (b) you breach any terms of this Agreement (including any Subscription Terms or Rules of Use).

In past, the TF2 "Announcer" took the items that were obtained through hacked achievements back for one week, which shows that Valve doesn't accept them. This also shows that it is not authorized by Valve.

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.