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3) I am not clear on what a Studer vent would do here, although I don't see any sort of venting going on under the sink with this drain as it is currently (I also looked under the house today and didn't identify any sort of venting system). Can you clarify as to what a Studer vent would do here and where should it go (underneath the sink? Under the house?)
Look on the roof above the kitchen. If you can see a pipe exiting the roof line then the sinks's vented. If you don't see a pipe get back and we'll discuss Studor Vents.