Do The Events Of Pacific Rim and Jurassic World In The Same Cinematic Universe As Godzilla?

No.

First, understand that Legendary Pictures can not fully fund or distribute the movies they make. They are partnered with larger studios that distribute movies for them.

In 2013 they were partnered with Warner Brothers which distributed Pacific Rim. In 2017 they were partnered with Universal who distributed Pacific Rim: Uprising. 

They helped Universal fund the first two Jurassic World movies, but unlike Pacific Rim, Jurassic World is a Universal production. That's the only reason their name is attached to the Jurassic World films.

Legendary has made a deal to go back to Warner Brothers, so next year WB will be distributing all of their films. It also means they will not be attached to the third Jurassic World in any way.

As for connecting with Godzilla, it's been stated multiple times by filmmakers from Pacific Rim: Uprising that the Monsterverse and Pacific Rim-verse do not connect, which is pretty obvious since the dates clash. The kaiju cameos in Uprising were just Easter eggs. Nothing to take seriously.

As impressive as the dinosaurs are in Jurassic Park, I doubt they would stand a chance against Godzilla, so even on logical grounds, it is not probable for them to exist in the same universe. 

Although there is no legally-binding connection, it's not hard to have the events of the JP movies take place in the same continuity as the Monsterverse.

There's really nothing contradicting anything, so if you want to hold that headcanon, nothing official is stopping you from saying it can't work in terms of chronology and what-not :P

Except that Jurassic Park is more grounded than any Godzilla movie. I mean Jurassic Park is a very hard sci-fi and Godzilla is so soft on the sci-fi scale that its rubbing belly buttons with fantasy.

Xenotaris: Exactly. If you are strict, a creature the size of Godzilla is scientifically impossible on Earth. Jurrasic Park relies on fringe science, but it has just enough believability that someone could not dismiss it 100%. (Of course, just because something is theoretically possible does not mean it is likely or possible in reality.)

So essentially, we are dealing with two different set of rules. Godzilla is more in the fantasy/pure make-belief genre while Jurassic Park is fringe science. You can say they are similiar, but they are not the same. 

Just to entertain the idea that would also make the upcoming Alita Battle Angel movie part of the same universe which would be... very interesting if nothing else. But unfortunately the movies being different studios kind of messes that idea up. Would be interesting though to see Godzilla vs the Jagers. Though I'm pretty sure the fight would be over pretty quick and Godzilla would be the clear victor

For the connection of Jurassic World to Godzilla: If the dinosaurs can't even survive a single volcano, then what makes you think they could survive an Earth full of giant monsters that are essentially living, breathing, radioactive natural disasters that can move wherever they please? Same thing for the connection of Jurassic World to Pacific Rim.

For the connection of Pacific Rim to Godzilla: The Jaegers couldn't even handle the kaiju coming from the breach that much, so what chance would they have against that and an Earth full of native kaiju, with both appearing at the same time?

Sorry, but I think it's safe to say that they are not connected in any way.