Yes i drain line has got to be vented. A sewer line creates gases which has to be vented otherwise two things happen. Either, you get terrible smells around your property or your sewer lines can explode due to trapped sewer gases, this is costly and can be a painful smelly experience.
It will usually be vented through the main vent.
All toilets have to be tied to a vent,
Vent piping is a pipe that runs outside and is connected to the drain line of your home. this helps the water in the drain to move to your septic/sewer.
Take your snake on the roof and clean the drain from the vent.
Install an automatic air vent on the line
The drain pipe should have a vent that goes up where the sink drain goes into it. That is where it vents. Sinks drain because there is nothing causing a vacuum when the water leaves the sink.
as long as it is not installed on a drain line. If it is used strictly on the vent system then yes it can be used horizonally.
Partially clogged drain line.
No plumbing code allows a loop in a vent line. They must be pitched just like a drain line as there are many conditions where water can be in a vent line, such as rain through a roof vent and condensation. If the "u" were to fill with water it would no longer function as a vent.
Drain line starting to clog or vent not working properly.
Make sure the drain line and the vent are cleared. Noises like that are usually caused by either restricted flow or a plugged drain vent.
Your drain line is plugged somewhere after your shower and toilet line are connected. The reason for the gurgle in your toilet is because air is trapped in the drain and your toilet is the closest place for air to flow to. If you had a vent on your toilet trap arm then it would gurgle in the vent rather in your toilet but your drain is still plugged and needs to be snaked out and maybe your drain line needs to be examined for right grade or broken.
There is an overflow pipe on every tub. There is no further need to vent.
Heck no as the sink is higher then the basement drain and water does NOT flow up hill BUT it does seek its own level BUT you can connect a sink drain to a basement drain if you properly trap and vent the line