Can I have my OWN living trust and a JOINT living trust with my spouse, and put different properties in each? What are the pros and cons?

Yes, you could have both so long as you only put your separate property in the individual trust and don't have a prenuptial or postnuptial agreement that prevents you from doing so.   Whether you can amend the joint trust if your spouse passes away first would depend on the terms of that trust.   As for the pros and cons, it really depends on your assets, motivation in setting up a second trust, and overall concerns and goals.