Unlike the O2 sensors, which are emissions related, the TPMS is a safety issue mandated by the government. I cannot imagine a vehicle passing a safety inspection with a TPMS code set.
Depends on the state. You can check pressure yourself, some states don't care accordingly.
In California it's a gray area. Smog checks are supposed to fail if the check engine light is on, since it's an environmental check. But TPMS doesn't throw check engine. It throws a DTC, which is a gray area as to if it's an automatic fail.
Personally, I wouldn't risk it. I've averted losing tires because I was at speed on the highway when a leak started. It's not worth losing a rim, plus you get rebalanced at the same time.