A few things to address here.
1. Depending on how new the Viper system is that they installed in your car, you may or may not have to press the start button upon entry. On my FX I unlock the doors, get in, press the start button twice (to engage ignition ON), then I step on the brake to disengage the remote start. I've had mine for about two/three years now (older model Viper). My wife's is like JD's she does not have to press the start button twice but she needs to have the factory key fob on her and press the brake to disengage the remote start (newer model Viper).
2. If my doors are unlocked when I remote start start the car from my phone they lock right before the car starts (I tried it). I've had Viper remote starts on numerous vehicles and they all locked the doors if they were unlocked upon receiving the Smart Start command.
3. Your Viper should be set to arm on it's own after a minute or so anyway, so there should never be an instance when the car is remote started and the alarm is not armed. Mine is set to arm but not lock the doors after a minute or so. I had it purposely set not to lock the doors so I never have the accident of locking the keys in but out of the box it should be set to arm and lock doors after a minute or so on it's own if it doesn't receive the lock command from you.
All that being said if you can step on the brake and drive off without the key fob on you then there is a problem with the installation. Sounds like the bypass they installed isn't doing it's job. If the Viper isn't locking the doors when they aren't locked upon receiving a Smart Start command and you can drive off without the key fob, it sounds like your bypass is not programmed correctly. A screwy bypass can give you all those problems and then some. The bypass is what tells the Viper how to work with your car, it's just as important as the actual Viper system that the bypass is functioning properly.