If the cop already knows you (it sounds like he did) and knows that you are on felony probation, then you can be stopped and searched without cause. Be very careful.
It depends. Do the terms and conditions of your felony probation require you to submit to warrantless searches? Then the answer is yes... unless the searches are conducted in an arbitrary or harassing manner.
Since your wife's purse was in the car and under your control, they could search it, too. If she was present, the answer might be different.
It sounds like you're a little unclear about the terms of your probation. I'd suggest reviewing them with your lawyer or probation officer.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like this site, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
Yes you can be pulled over and searched without cause or suspicion. That's because in agreeing to accept the benefits of probation you agreed to give up your 4th amendment rights protecting you from unreasonable search and seizure.