Sounds like you have a positive air pressure situation. If the vent under the sink is an "AAV" or a Studor vent then it would only alleviate a negative pressure situation. When you have a plug of water going down the drain the vent, if properly ran, would allow positive pressure (air in front of the water) to escape and then allow negative pressure ( air behind the water) to come in and would balance the pressure in the plumbing pipes. When you run your sink the air has no place to go. Depending on how the vent was pulled off of the toilet and how the sink waste line was tied into the toilet drain can also be a factor since the you'd think the positve air would exit out of the toilet vent, which should be the case. The fact that the toilet does not flush tells me that the vent is not installed correctly. So all of this positive air pressure has to have somewhere to go and that means out of your toilet, hence the bubbling and out of your shower which explains the splashing of the trap water as the air escapes. I have a feeling that your shower may not be vented correctly if not at all. I would check to see if the shower has a vent and if the toilet has a vent installed correctly. If the vents on the toilet and the shower are ran correctly then the Studor vent under the sink should be ok for the application but I strongly advise running a conventional vent for the sinks. If the plumbing system has a venting problem this causes the solids to drain very slowly and can cause the line to back up which would make all of the other problems worse. And of course I have to ask the obvious, has the septic tank been pumped?