Barcelona from Las Ramblas to Beach (the dock where the tourist stuff is) by bus or metro

You walk. Seriously.
Start at the top of Las Ramblas and, holding tightly to your wallet/purse, you walk down to the marina. Along the way you can dart in and out of any number of interesting sites. At the bottom of Las Ramblas is the columbus monument. Access to the cable cars is a short walk. However, most of the remainder of your wish list involves the subway and walking.
I recommend you open up google maps and plot the location of the items on your wish list and then establish a priority for visiting.

This will be a very full day. It sounds like you'll start with your top walking priorities, then go up to Montjuic by afternoon. The museums on Montjuic and many other sites are open until 20:00, and people are strolling the Ramblas and gothic quarter at all hours, but the Boqueria market is more lively early in the day.

If you are an able bodied walker, you can easily do everything on foot. The beaches and boardwalk (Gehry fish) are a bit of a hike from the foot of Las Ramblas (Columbus monument) but you want that because of the views. Then backtrack some and take the cable car up to Montjuic where you can see the Miro Museum and then walk down to the Mies Pavillion and the art museums. Once done there, catch the metro at Espanya back to wherever you want. PS - There is a great Miro sculpture just northwest of Espanya between it and Sants Station. You can't miss it.

You'll fill the entire day if you see and do everything on your list, but some aren't really worth much time and you can decide what most interests you or not.

The Ramblas start at Placa Catalunya and end at the Christopher Columbus Monument - which is where the dock area you're talking about is located. The cable car leaves from these docks (maybe do this last after you walk up the beach). I was there in April and there wasn't much happening at Montjuic so I never went there or rode the cable car.

On your left the entire Ramblas is the Gothic Quarter (assuming you are walking toward the ocean).

When you get to the end of the Ramblas (facing the ocean), turn left on the boardwalk and you will walk past the beaches - each beach is man made and small. It's walkable to the Olympic Village (where the fish statue is).

This is a lot of walking for one day, particularly if you want to take in some of the sights along the way - depending on your walking ability, the stuff you want to see, and the pace you like to sight-see at, you may be better off planning two days for all this.