The grey overflow pipe looks as if it has a couple of near horizontal sections, where, if you are in the habit of filling the bath quite full, a little water might overflow and settle in the grooves of the grey pipe. Any odour developing in the overflow is not protected by the trap, so with the plug up you might get a smell, or from the overflow aperture itself.
You might try flushing water through the overflow, or putting a bit of bleach down it?
If it was me installing the overflow, I would have cut the pipe shorter and made sure the run was more vertical from the overflow fitting on the bath down to the trap.
When you empty the basin, the slug of water wooshing down the drain system pulls air in behind it, and not enough air seems to be able to get through the basin trap, so some is pulled through the bath trap and you get that gurgling. This might indicate the basin trap is a bit gunged up, so taking it to bits and cleaning it, or drain cleaner in the trap, might help
