Did the writers forget that Vito and Adriana are cousins?

Good Catch.

Adriana's mother was Richie's sister, but how was Vito linked to Richie? Were they blood-related?

Good observation but not necessarily poor writing there. Often times on the show characters who were cited as cousins or nephews weren't directly related by blood. Tony and Christopher are an example. Christopher was in fact Carmela's cousin yet Tony took an uncle/mentor role.
In Adriana and Vito's case it can easily be argued that they weren't related by blood, and were either cousins by marriage or distant cousins. As Adriana told her FBI handler: "It's an Italian thing".

An example of sloppy continuity I found was when Vito and Phil were made close cousins in the 6th season. Yet there was never any indication of this in the previous season, and during their construction yard meeting they did not appear remotely close, while Vito seemed to be meeting Phil's brother for the first time.

Not poor writing. In Italian families everyone is loosely referred to as either cousins or uncles. It's a gag the show gets real tongue in cheek with.

"Aren't we cousins or something."
"Ah it's so far removed now Tanto's a closer relative to you."

"Cousin" could mean a third in-law's second in-law's distant relative. Point is: never take that word seriously when an Italian says it.