Hi Melissa,
Congrats on the triplex. Before I give you my answers, I want to say that I am not a lawyer or tax advisor - I just have some knowledge that may help you. So, here we go:
1) I would not hold the property in your name. It's easy enough to set up an LLC or LP. But when you do, do not incorporate in California. Nevada or Wyoming are the only two States to consider. There is a saying I alway remember when considering California laws: "Go to California on vacation; come back on probation."
Meaning, there are so many laws in your State, it is easy to get caught up without intending on doing anything wrong. Plus, there have been several court cases concerning corporate protections in California where the Court has decided to "pierce the corporate veil" concerning California LLC's - which means that setting up an asset protection structure based in California is almost like not having anything in place at all.
The thing you have to keep in mind is, right now, you are completely exposed. If one of your tenants falls down the stairs, they will go after you and all your assets. If you set this up correctly, your assets will be protected.
My suggestion - pick up Garret Sutton's book: How to use LLC's and LP's - here's a link: https://www.amazon.com/How-Limited-Liability-Compa... He has a chapter specifically for real estate - this will get you to understand what options you have and why putting something in place is so important.
2) Yes, once you set up an LLC or LP to hold the property, I would set up a business bank account and deposit all rent checks, plus pay out all expenses from that account. It makes it much easier come tax time.
3) If you do the first two things, you will be able to deduct all business expenses.
4) I'm not familiar with anyone in California with regards to a tax advisor, but for a legal advisor - I would consider Garrett Sutton's company, Corporate Direct. Just let them know your situation and they will be able to advise you as the best structure to set up to meet your investing needs. Here's a link: https://www.corporatedirect.com
Hope this helps....