Could a slow flushing toilet & smelly shower drain be a clogged vent stack?

very much so. The toilet is probably on the same line further down from the shower. When it flushes it sucks the water out of the shower trap and then sewer gasses can get in.

very unlikely because that vent is 3 or 4 inches inside diameter.The main line under your house is 4inches schedule 40(thick) and all the other pipes feed into it, the toilet is the only 4" fixture. under every fixture is a p-trap that holds water to prevent smells and gasses from coming into the home, since the toilet holds water, it is it's own p-trap. It is very possible that there is not a p-trap under the shower pan and if your house if off the ground, you can get under there and check, and also there are some pipes partially stopped up. Unstop all lines first and i prefer a mechanical way, not chemicals. I worked at The home Depot an they have some excellent books that you can look at and then buy if you like, that's how i would show a customer what i'm trying to tell you, good luck cause plumbers are expensive.

From my years of smoking in the jacks in school. The cigarettes never clogged the toilets. And I say there was at least 100 flushed down each day