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Most bathroom drain slowdowns and blockages start at the popup assembly, not at the trap or in the other drain lines. First, a brief description of the popup assembly. The stopper in your sink moves up and down by means of a metal rod that extends into the drain pipe right beneath the sink. This rod pivots when the popup handle is moved, raising and lowering the stopper.
The culprit is the metal rod that moves the pop-up up and down. Because it extends into the "line of fire" of the waste leaving the sink, it collect bits of hair over time. This mass of stuff grows ever larger until it eventually slows down sink drainage or causes a full-blown blockage. Unfortunately, drain cleaners will not dislodge this stuff… they can make it even more stinky! Partial disassembly of the popup mechanism may be necessary to remove the stuff.
The first step is to extract the stopper from the sink. If it pulls right out, you are in luck. Sometimes you can use a long screwdriver to push the blockage through. Then simply flush the drain with a few gallons of very hot water.
However, if the stopper does not just pull out with a gentle tug, it is interlocked with the popup mechanism. You will have to unscrew the popup rod under the sink, which is holding it in place. If you look underneath the sink with a flashlight, you will see the linkage that operates the stopper. Unscrew the large nut on the side of the popup assembly (you may need pliers or a wrench), and pull out the rod, which will free up the stopper. It is alright to bend the linkage slightly to do this... just don't bend it any more than necessary to release the popup. When you are done you can bend it back.
(A note… when disassembling the popup mechanism, don't lose any of the small washers! They must be reassembled in the original order and direction or you will have leakage around the rod and nut.)
As I said, you will probably find lots of gunk on the stopper and the metal rod that drain cleaners will not remove. In fact, they seem to make it even more stinky! Pull out or push through what you can from the drain and clean the popup. If the blockage is too thick, you might have to disassemble the trap to get it all out.. The trap is an "elbow" under the sink that holds a small amount of water all the time. This standing water acts to block sewer gas from seeping into your home through the drains.
After reassembly of the popup mechanism, apply some lubricant, such as plumbers grease, to the threads, seals and the pivot ball on the popup rod. It will improve the action of the popup and make the next disassembly easier.
Flush the pipes with a few gallons of very hot water to be sure all matter has washed through to the larger drain pipes beyond the sink drain. Please… don't forget to reinstall the nut under the sink first, or the water will drain inside the cabinet!
As a "prophylactic" against future blockages at the popup, try to get into the habit of filling and draining the sink with very hot water every week or so. Due to the proliferation of low-flow faucets, most bathroom sinks have only low volumes of water running through them. This can cause the collection of all sort of stuff in the pipes… especially if you have a long sloped drain pipe running from the sink. This boost to the flow may help!
In a properly plumbed house, water should be in all your traps all the time preventing the entrance of sewer gas into your home. Traps, as you may know, are bends in the pipes that hold a small amount of water to block the entrance of sewer gas into the home. Look under any sink and notice the s-shaped bend in the drain pipe.
In plumbing design, care must be taken when installing multiple plumbing fixtures on the same drain pipe or else the movement of water from one fixture can literally suck the water out of the traps in the others. For example… if a toilet is on the same drain pipe as a sink, flushing the toilet will empty the sink trap in an instant. This loss of trap water can also occur in a single fixture if the distance from the sink toilet is too far from the larger main pipe it feeds into. Again, the result is the same... the water leaving the sink causes a vacuum and sucks the trap dry. Generally this would not occur in a single, long waste line unless the sink were filled with water and suddenly drained. The relative trickle from a typical sink faucet would probably not establish enough suction to empty the trap.
This problem is addressed by the use of plumbing vents which allow air into the waste lines "behind" the moving water. These vents can be physical pipes venting to the outside… usually the roof… or mechanical valves that selectively open to admit air into the system. This extra air prevents the formation of a vacuum behind the moving water and thus keeps the water from being pulled from the other traps. Consequently, blockage in the venting system or poor design can cause traps to empty and sewer gas to enter your home.
If the toilet seal was leaking gas, it would also most likely be leaking water. Check the floor around the toilets for evidence of leakage. If you have a vinyl or linoleum floor, leakage can be hidden for years between the flooring and the subfloor… especially if it is just a slight seepage. (Be mindful that this seepage can also generate odors.)
After you flush the toilet, does the water in the bowl rise at least a few inches above the level of the bowl's drain? This is important because the water in the bowl acts as a "trap" to keep the sewer gas out. It is possible that some gas could escape if the level in the bowl is too low. If you have the tank water level set too low or are using water displacement devices in the tank, such as bottles, bricks or "commercial" products, the amount of water entering the bowl will likewise decrease.
Another possibility is that you are getting downdrafts off the roof and the odorous exhaust from the roof plumbing vent is entering through open windows. Since the windows are probably closed most of the time in the colder months, you wouldn't notice the problem in the winter. Downdrafts are often caused by tall trees too close to the house. Cutting down the trees or increasing the height of the vent "stack" (the vent pipe that exits the roof) might help if you determine this to be the problem.
There is the remote chance that there is a trap missing in the tub/shower. When the seasonal odor appears, tape a plastic bag over the tub/shower drain and see if the odor dissipates. In fact, you can do this to each fixture to try and isolate the source of the odor.
I am running out of ideas... but I do have one, final thought. It may be possible that the odor is not sewer gas at all, but instead accumulation of gunk within the sink drains themselves. There can be quite a bit of gunk collected above the sink trap that gives off a rank odor as it decomposes. This would be more obvious in the warmer months since heat speeds rot... ever notice how the kitchen garbage has to be put out more often in the summer? And further away from the house, to boot?
This "glop" can be removed through the occasional use of a liquid drain cleaner, but can also be cleaned out manually by disassembling the trap and just scraping it out. Remove the pop-up drain stopper and look in with a flashlight first... if there is absolutely nothing there you probably don't need to disassemble it.
While you have the drain trap apart, don't worry too much about the stuff on the "downhill" side of the trap unless it seems to be causing a slight blockage... the water in the trap will keep that smell away.
One thing about liquid drain cleaners is that they can actually increase this odor problem by accelerating the decomposition of the gunk while not quite removing it from the pipes!