Can vermin come in through your toilet or shower drain and, if so, what can you do about it?
The smell of sewage is definitely a bad sign and means that either a trap has gone dry or a connection somewhere is bad releasing sewer gasses.  Maybe this is related to the vermin, maybe not, but its definitely something that needs a resolution.

When checking traps, check:


Are all your toilets full of water?
Do you have any sinks or fixtures that are very infrequently used (eg: basement laundry sink)? 
Do you have drain and trap in your basement floor? Does it have water in it? Is there a trap primer hooked up to it (this is a small pipe that is connected to a frequently used fixtured so that when you run the fixture, a small amount of water flows into the trap and keeps it full)


Other things to check:


Do you have a sewer clean-out inside your house? If so, is the cap done up tightly? Also check the gasket on it - they can dry up and start leaking sewer gasses due to the poor seal.
Bottom of toilets - flush each one a couple times and look for small amounts of water near the base. This would indicate a failed wax ring.  Make sure to flush a number of times because if the leak is very small, it might not be noticable after a single flush. 
Vents on roof - are any drain vents near air intakes (sucking the sewer gasses back in to your house)
Signs of a sewer line blockage - gurgling sinks, slow draining fixtures, sewage coming out of the lowest drain in your house (often the floor drain in your lowest level)


I had a situation in my house where one of the connections in the clay sewer had sunk by an inch or so.  This would cause stuff like toilet paper to start collecting resulting in a partial blockage. Overtime, it would get more blocked until finally the pressure of the water combined with atmospheric pressure would cause it to push through. During this process, it would suck all of the water out of my basement floor trap and I'd go from a full trap to a completely empty trap very suddenly.  Another interesting thing here was that when running a camera through my sewer, I noticed spider webs in them, so that definitely seems to indicate that some critters might come up from the actual sewer itself.

Ultimately my point here is that it can be a lot of things and if you can't find the cause on your own it is advisable to call a plumber - the sewer gasses are not something you want to be breathing in all day.