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1) The bar sink and the basement washer each need their own vent, right?
Every fixture that has a trap must be vented. You have already addressed this,(see image)
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to the vents should be located somewhere between the soil stack and the fixture's p-trap, right?
On a 2" line the vent must be located within a 5 foot limit between trap and stack vent.
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3) The vents from the basement fixtures should tie into the stack somewhere above where the new, main level washer ties in, right? Does it matter how high above they tie in?
They should revent back into a existing dry vent at least 6 inches over the highest fixtures flood rim. It doesn't matter how high above.
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4) What size pipe should I use for the two vents?
I would use 2" unless there's a problem with size.. In that case drop to 1 1/2".
Good luck and thank you for rating my answer. Tom