This question or some variant of it seems to come up frequently on Avvo. In general the answer is yes but for this answer to be definitive, we'd need a more complete statement of facts. In your automobile question, it would be unusual for officers in a routine traffic stop to ask if anyone in the car was on probation. If they did and got an affirmative answer they'd be entitled to search the person on probation and would probably be justified in searching the other occupants and visible portions of the interior to insure their own safety.
My response to the second question is similar. The cops can't just knock on a door and ask if anybody is on probation. But if they have some reason to believe someone inside is on probation they might enter and look around. Whether this would stand up to 4th amendment attack is another question but the only person with standing to challenge the search would be the homeowner or lessee, not the probationer.
See also my guide on probation searches at http://www.avvo.com/legal-guides/ugc/probation-searches-some-rules.
Maybe. In the auto scenario, they could definitely search the person on probation and may be able to conduct a "Terry" frisk for weapons and anything in "plain view" that might provide probable cause for a further search but, searching the passengers or the vehicle beyond this would probably be subject to a motion to suppress if anything was found.