The Hatake clan is not originally from Konoha, they are from the Land of Iron. Most of them are farmers. They adapted to the cold snowy winters and warm summers of the land and thrived (they tend to do more harvesting than husbandry and it's not always that cold, I just liked the snow vibes for the drawings).

With time some members of the clan deviated from farming and trained to fight as samurai, so eventually they were established as a side branch.

A few of those samurai learned ninjutsu and became shinobi (I think it was mentioned in the manga that that is rather common) and left the land. A young Sakumo was among a group of shinobi Hatake who travelled to Konoha looking for a new life.

After arriving there and going through some messy papework and politics, they were accepted as a formal clan in Konoha (kind of like what the Uzumaki probably did). Sakumo married a Konoha woman and Kakashi was born. (Sakumo being a foreigner wasn’t an issue after witnessing his strength and loyalty but those prejudices resurfaced after his failed mission and had a big role in his depression).

Those few Hatake died during the wars and soon Sakumo and Kakashi were the only ones left. Kakashi had heard stories about the clan as a kid but he was too young when Sakumo died and forgot many details such as them not being originally from Konoha. As far as Kakashi knows he's the last Hatake (which makes the idea of him finding out about his clan very fun to explore).

Some other cultural ideas:

The Hatake usually have dogs to help them with the farming duties, which means they commonly have shepperd dogs, the main one being the Samoyed dog (big fluffy white dog).

The samurai branch, however, use wolves instead since they are better for combat. This transaltes into Sakumo having wolf summons.

They have a close connection with nature and prefer living separated from big cities or populated areas. They also value chosen family over blood family.

I'm not a big fan of the Hatake having dog/wolf attributes like the Inuzuka. They treat their animals as part of the family and respect them but they don’t have physical or instictual features. As for Kakashi’s sense of smell and fangs (a hc I like), I'm more inclined to the idea that his mother was related to the Inuzuka or mainly that they're just a Hatake genetic attribute with no other implications (like Zabuza's teeth or other genetic mutations born from the shinobi world).

I like the concept that Kakashi feels comforted after learning about his origins because his clan nurtures life instead of taking it. Not every aspect of his life is tainted by blood.

The Hatake generally have stark and leathery faces, but Kakashi takes after his mother and has more delicate features so he doesn't look like a standard Hatake aside from the white/silver hair.