Do not be too quick to dismiss your previous plumber for suggesting a AAV (Air Admittance Valve, "studor vent") for your kitchen island. Although the Uniform Plumbing code(UPC) & International Residential Code(IRC)have radically differing views on how a kitchen island vent is to be constructed, nonetheless, both the IRC & UPC do allow an AAV for kitchen island venting. Under the IRC all structures are required to have one "Main Vent",which must run undiminished in size from the structure "Main Drain" through the roof. After that we are permitted to install an unlimited number of AAV's. Under the UPC they do not require a "Main Vent" but all fixture vents are required to be sized by the total DFU (Drainage fixture unit) load of all fixtures served by that vent. Under the UPC a structure is only permitted a maximum of one AAV, with the expressed written consent of the local inspecting authority, and in most cases a kitchen island is the only location where they will grant the exception. Now what is the difference between the kitchen drain and your bathroom drain you ask? When we have a sink on an island it is generally the only fixture attached to that line. As the water drains from the sink it pushes the air ahead of the water in the pipe. Under some circumstances this can result in a slight vacuum in the pipe behind the moving water. This partial vacuum would then result in the standing water in the trap being sucked into the pipe by siphon action and once the water has passed on down the trap is left dry, allowing sewer gas to enter the structure. In this case we install an AAV, which would permit air to enter the pipe behind the liquid flow, thus negating the suction effect. There is a slim chance that you might resolve your problem by installing an AAV on the lavatory drain line, an when you consider the cost of installing an AAV as opposed to opening walls it might be well worth your time to try it, however I suspect you have a different problem in your bathroom. You state that you hear a gurgling sound in the tub or lavatory when another drain is used. It is possible that the water in the gurgling trap is being sucked out, and in that case an AAV might resolve the problem, but I suspect you problem is not a result of negative pressure (the partial vacuum) but rather it result from positive pressure. As the water from one fixture flows through the pipe it pushes the air ahead, causing a slight positive air pressure. Normally that positive pressure would be pushed up and out a roof vent however if your system does not have the required roof vent as one fixture is draining it will push that air up through one of the adjacent fixture drains and out through the trap. Keep in mind that the air in the line contains sewer gas so as that air & sewer gas mix is pushed out you get the odor. The odor may seem greater at some times than others. That does not mean the sewer gas is not leaking because not all the component of sewer gas are odorous. In fact, one of the main gases that comprise sewer gas is totally odorless.

I recommend that you pick up a copy of a good basic plumbing book, which will provide you with the general information that you are seeking. Once such book is the Black & Decker's book entitled "Home Plumbing", and it give all the charts and tables that you need to figure out what size drains to use, how large the vents need to be, where they need to be located, etc. The biggest challenge with a project like this is the venting. If you use the main stack as the vent, you can't tie your new vent into the stack until at least six inches above the flood rim elevation of the highest fixture that discharges to the stack (typically a sink o the second floor of a 2-story house).

Be sure you understand the options. There are 2 kinds of crawlspaces allowed by the IRC: Vented and Unvented (aka "sealed"). The terms "unvented" and "sealed" are misleading because this type of crawlspace is not completely sealed or unventilated. The IRC code requires that both of these types of crawlspaces be "ventilated" so that moisture does not collect and encourage the growth of fungus (whether from the ground, condensation on the slab, etc.) by one of the following methods: 1) "Vented" Crawlspace - install vent openings in the exterior wall of the crawlspace 2) "Unvented" Crawlspace - inside the house supply to or exhaust from the crawlspace a small amount of air with a return air path to or from the house and insulate the exterior walls of the crawlspace. Both crawlspaces must have a vapor retarder at ground level and an access opening. For more information see IRC section R408.1, 2, & 3.

Sure...it isn't "easy". It takes finesse to get in a porcelain tile floor (and YES...I'm still recommending PORCELAIN for a basement). You said you wanted easy. Vinyl planks or vinyl click-lock flooring = easy = SUPER easy in a basement. Everything else gets harder and requires plenty of skill/knowledge/training to get into place. And there are plenty of high-end vinyl planks/floating tiles that look MAGNIFICENT. The "cheap" vinyl you are thinking of is still out there - but they are easy to spot which means they should be very easy to avoid. The HIGH END vinyl products that are extremely realistic = $4-$7 per square foot. They are not cheap. They do not LOOK cheap. But they are STUPID-CHEAP/easy to install. If you are confident in your ability to test for and then mitigate moisture in your slab to get a porcelain tile into place, you are welcome to do so. I do NOT recommend ceramic. It is a heart ache waiting to happen in a basement. Good luck. Let us know your skill/training level. Perhaps we can offer up a few more options.