The first two are just continuity errors. It sucks but it happens. Pretend it happened 5 minutes before what we saw in the finale. They get there, see the suit and think it's a waiter. Nobody knew yet that Veronica had been there, assuming maybe she was staying with Shapiro to discuss the police stuff and she'd been on set all day. They Hebrew their conversation, walk off to meet with other cops arriving on scene, get more information and then the finale scene happens

For the time, assume there was a lot of downtime in between episodes unless stated otherwise. They only had a handful of interactions with Veronica and Shapiro and there's no reason they all had to happen immediately after each other. We didn't need to see the boring, unrelated stuff that happened on the other days.

Other people were involved in Longwater so it's possible someone found out and decided to blackmail the people involved but stealing the disc. It was also a sex tape so it may have actually just been about that. How they found out is a mystery but not a plot hole or anything. The lady at the beginning could have just been a woman being paid to hire Archer and the staff just used an actress they're friends with to supply the voice.

Veronica not remembering Archer may have just been her lying so that people didn't know she was associating with private investigators. She did that in the Bel Panto episode.

The dialogue by the pool was supposed to be poor. Archer was clumsily trying to get Veronica to confess and she was trying to manipulate him and thought he was an idiot. He made a big deal about the gun because he wanted Veronica to shoot him to prove she was a killer.

I think other people have answered the other questions. Maybe some of that helps. I liked this season and I'm really hoping we get another.