The baseball bat is for "Attitude adjustment", apparently :)
Source: Arstechnica photos of ISS control room
(ADCO=Attitude Determination and Control officer)
Here's a better pic of the bat from the Twitter:
Yes and no. The current X-37 as flown by the US Air Force is not equipped with any docking ports (PMA, CBM or APAS) as described by @geoffc, nor does it have an attachment for the SSRMS (aka Canadarm) to grab it.
However, Boeing, the manufacturer, have published designs that would permit it to be so equipped at some future time. These plans are described in ...
The main 'throat' of the PMA is not slanted -- it is a 31.5 inch passageway that is straight as an arrow. However, the throat has to have a bit more open room on one side because the hatch of the Russian docking port opens outwards. If there was not a space for the hatch to open beyond 90 degrees, it would block the main passageway.
The NASA document Space ...
The drawing in user10795's answer is not an accurate design drawing of the PMA design and thus the conclusions flawed.
In this document the ISS redesign history is discussed and the PMA design existed at least as early as 1993 in fig.10 the Russians propose a first design of ISS combining Russian and US modules, the PMA designed by Boeing can be seen on the ...
The ISS has 3 basic docking port types.
2 PMA's - used by Shuttle. With addition of LIDS adapter (coming on SpaceX CRS-7 in June 2015) the CST-100 and Dragon Crew will dock to it.
2 CBM ports - HTV, Dragon Cargo, Cygnus, and ISS modules berth to these.
4 APAS ports - Russian segment has 4 ports. ATV, Soyuz, and Progress use these.
You can see a description ...
Forget ISS and the Russians. Its the legacy design, inherited from Boeing's Pressurized Docking Adapter on Space Station Freedom. Originally, one of a pair next to each other on the end of Hab and Science modules (or nodes in earlier design), to fulfil the spec that two Orbiter-spec docks be available.
Okay, long story short:
The fact that concept ...
The slant is to allow big payloads to be taken out with adequate clearance:
Meanwhile, the ISS design had to accommodate the shuttle. It needed to provide a zigzag tunnel mechanism (the Pressurized Mating Adapter) to optimize the clearance to remove payloads from the bay after the shuttle had docked.
— Hale, Wayne, and Helen Woods Lane, eds. Wings in Orbit: ...
This is IMO a great question. I was a bit surprised to see "soft-berthing" used to describe a berthing system that can't possibly have anything in common with using rubber buckling or foam filled fenders when mooring ships, or even airbags on landers for cushioned, soft-berthing and/or landing. Well, I guess any new designs have a bit of a leeway with naming ...
They would most likely use a TLA (Three Letter Acronym) instead of the noun "port". In the example you give, they would say the "port CBM" (Common Berthing Mechanism).
Here's an example from the STS-130 Flight Plan
Maybe they say it here on page 81
“Pressurized Mating Adapters
The Pressurized Mating Adapter was a transition structure originally designed to provide the interface between the Space Shuttle Orbiter and the Space Station Freedom. Its unique shape was developed to preclude contact ...
The simplest method of transfer to the rotation wheel is similar to what is shown in the movie 2001: A Space Odyssey. The rotating ring is within a non-rotating outer hull. One side of the hub is an open ring through a bushing and into a tube; the other side is a central axle. Near by, behind the axle side, is the counterrotating flywheel.
The tube entry ...