I have seen stand up showers installed without P traps, although your "new" home should have one. The "flapper" valve is not what's important here, rather, having a functioning p trap at every sewer feed is. You have to keep an air block.
peppermint test, hmm, that sounds interesting.
The vent stacks are to allow the waste lines to drain properly without sucking the water out of your p traps. I suppose it is possible for a blocked vent to prevent your shower ptrap from functioning properly. The best way to check your vents for blockages is to take a garden hose up on the roof and poor a LOT of water down the vent. You should know pretty fast whether it is blocked. It is also possible that the ptrap on the shower is not installed in a way that a vent is doing it any good. (all the vents work, but there isn't proper venting for that one ptrap.)
This will be a lot easier to fix if you have access to the area beneath the shower.
Call a really good plumber, (get some recommendations from some large remodeling contractors in your area)
If your home is 7 months old, your builder should be responsible for fixing this, assuming he is honoring his work and warranties. That can be a challenge in itself.