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Yes a clogged VENT pipe can cause a sluggish drain
Every building has a "stink pipe." Usually located on the roof these pipes vent air to allow the plumbing drains inside the building to work correctly. As water flows into a plumbing drain pipe, air is vented out of the drain through the "stink pipe." The pipe that goes from the bottom of the back of your toilet, up the side of your house to let any gases produced by the waste escape into the air instead of bubbling back up through your toilet and sticking up your house.
Rodents can and do get into almost any pipe.
No
By looking for the vapor in colder areas
Every building has a vent terminal in modern societies
put it on ya stink pipe
Vent, return circulation
Call a pumer
You can extend the faucet pipe by, getting an extra metal pipe and welding it with the faucet pipe and it shall be extended.
Bathroom sinks use 1 1/4 pipe for the downspout in the sink. You can use 1 1/2 from that by using an adapter washer in the connection above the trap.
There should be a nut around the chrome pipe holding it to the brass. Loosen the nut and pull the chrome pipe out.