Give it some time and more knowledgeable people will come and answer. I know a red-taged item cannot be put in service. It would be against the law for someone to turn on your fan, or to hook up electrical service to it if it were not hooked up already. For a professional, it would likely be a licensing issue. For you, it is an outstanding violation that probably means you cannot sell your house. There is essentially a lien against your title if you continue to ignore it. These are all my guesses, based on things I have read. I googled, "What happens if I don't fix a red-tag violation in my house" and found one town said this: "Red tags must be recalled for inspection and paid, if applicable, within seven working (7) days. Failure of the contractors or owner/builders to request and receive re-inspections before proceeding with additional work may result in disciplinary action or a fine against the contractor or owner/builder." If you live in an incorporated city or village, go to the website, there is likely a FAQ with lots of info and maybe your question is answered. If you are not in an incorporated area, go to the county site - whatever governmental entity sent out the inspector. See if you can figure out the answer. Or just call the permit office and ask what happens when a red tag is ignored.

The general rule of thumb is that a bathroom exhaust fan should provide 4 complete air exchanges per hour. You begin by computing the total cubic feet in the room, then multiply that by 4. By example, your bathroom is 8'W x 10.5L which is 84sq.ft You would then multiply that by the ceiling height to get cubic feet in the room. For the purpose of this example I will assume an 8' ceiling, thus we get 84sq.ft x 8' height equals 672cubic feet. The total volume of the room in 672cu.ft and we want 4 air exchanges per hour so we would need to move 672cu.ft x 4 = 2688cu.ft per hour. The fans are rated in CFM(cubic feet/min) so we now divide the total cubic feet per hour by 60min/hr to get the required CFM to meet your need. 2688cu.ft/hr divided by 60min/hr = 44.8cfm. Once you know the theoretical CFM you select a fan that is rated as close as possible. In our example the theoretical CFM was 44.8cfm so a fan rated at 45 to 50cfm would do fine. P.S. if your ceiling is not 8' you will have to recompute the total cu.ft for the room but you should get the idea from the example.

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.

Decide carefully. Do you really need a bathroom TV? If you still decide to have one, consider first of it is water resistant, its mounting options, picture and sound quality, voltage, safety installation and warranty.