What it looks like is you shove the hose in, then pull the plastic clip outwards to lock it in. If water sprays out of it, you need new o-rings. If the hose shoots out, either that's not the clip, or some of the plastic is chipped making an unreliable lock.
If this isn't the case, could you tell us what you've done to try and fix it already and what moving parts you see? Perhaps a photo of the top of the faucet may help too
No, there should not ever be two traps on a single drain line.
The point of the trap is to create a water barrier which prevents sewer gasses from coming up the empty pipe and into the house.
When you have two traps on the same line, you end up creating a kind of vapor block in the line which prevents it from draining properly.
What do you do?
Get yourself some 1 1/2" PVC (not CPVC, which is yellow, PVC which is white) with a 90 deg elbow, and also get two Fernco couplings (black plastic couplings with straps on each end that can be tightened) that will fit the pvc and iron drain pipe.
Use a metal blade on a sawz-all and cut out the p-trap in your basement. (Have a bucket handy!) Leave room to connect the Fernco coupling on both sides. I'd cut the right hand side out just past the coupling.
Measure and cut the PVC so that with the 90 elbow the ends line up with the newly cut drain lines. Glue the PVC together and confirm the fit, then connect to the old drains with the Fernco.
Viola - a proper drain line that will drain smoothly w/ no vapor lock.
Best Answer
My off-the-cuff answer is 33" minimum but 36" highly recommended.
But the correct answer is: Get the installation manual for the sink. Usually available online, and if it isn't then contact the manufacturer. The installation manual will show you how much space is needed, particularly how much clearance is needed under each side of the sink, which will vary depending on how it is mounted. Keep in mind that some cabinets will have thicker walls and/or framing pieces under the left and right sides than others.
It is possible that a 33" cabinet might be 1" thick on each side and leave you with 31" of usable space. Combined with a 30" sink that needs no extra space underneath (i.e., it is drop-in and secured entirely on top), you would be just fine. On the other hand, a cabinet that ends up with less usable space (I wouldn't go down to exactly 30" for sure) or a sink that requires mounting brackets underneath the left and right edges will certainly need a wider cabinet.