You seem to have your seam-leaks under control.
As for the drain, there should NOT be a trap in the drain on a pop-up (PUP) unless it's a High-Wall. But the trap in the drain on a High-Wall also has its own vent. I can't tell what you have, but I'll presume a standard (low) wall PUP. In a standard wall PUP, however, a virtual trap can form when you flip over the sink from its stowed position into it's ready-to-use position. The flex hose that connects the sink drain to the outside port can "belly" downwards and form a trap of sorts. Venting outside won't help this situation, because the trap is upstream from the vent. Instead, you must make sure the drain hose is positioned so it does not form a belly in which water (and sediments and foot particles) can settle.
In other words, when no water is going down the drain, that hose should be full of air only, and when you pour water down the sink drain, then your sink drain is the vent, and the water flows reliably through the side port into the bucket or other capture device.
If your drain hose bellies and traps water, look at the design and determine why. It's not supposed to belly. Often there is a shelf or some other support for the hose so it stays level with or flows downhill to the outside port. Fiddle around a bit and experiment with small bungees or zip ties to support that hose at the point it sags and forms a trap. That should be there from the factory, but may have been overlooked. It's a simple fix to keep that hose from sagging. If I recall, on my first PUP, I added a tiny bungee that I unhooked for travel and hooked around the drain hose to hold it up when in service. Very easy fix.
Also, most people collect this grey water in a 5 gallon bucket. But, if you choose to connect your outside port to a sealed container, the CONTAINER must be vented to allow air out of the container as it fills. There are jugs that will accept a garden hose fitting into the jug, and that makes the jug a sealed container. All you need to do is remove the vent cap that needs to be removed when emptying the jug. Just remember to install the vent cap when carrying the full jug, or you'll get wet.
The drain vents are a good idea for some applications, but all drain vents must be UPSTREAM of any trap, so adding one of the pictured drain vents outside when the trap is in the drain hose inside won't help much if at all.