Hey John,
As an agent that works with investors, here is what I tell my clients:
1) If you don't have the rent roll already, request it. I highly recommend you tour the property with the property manager, observe the rapport that they have with the tenants and talk to the tenants directly about how they like the neighborhood. This is the best way to get a sense of what the community is like. Further, ask the owner what concessions (free rent, etc) were given to get the property leased up. These are typically never stated in the financials nor accounted for in the cap rate and yet may have been used to skew the financials.
2) Evictions depend in large part on the city you are buying in. For instance, evictions in San Francisco work differently than in Fairfield. Talk to the manager/owner about the process and ask how many evictions they typically do a year and for what reasons. You want a manager that is vigilant on keeping the riffraff out of the community.
3) As I understand it, you will need to honor the leases in place as they survive transfer of title. That said, if a tenant's lease has expired and they are on month to month, you should be able to negotiate new lease terms. Look at market vacancies for comparable properties to get a sense of what's really going on. If this property has vacancies way above or below market, put your guard up.
4) If you're talking about the security deposit from the tenants, that should transfer over at close of escrow.
In general I recommend getting as much due diligence as you can before getting into escrow and making your offer on those assumptions. If you're financing, I'd also involve your banker or financier as early in the due diligence process as possible, being careful not to throw every property you're casually considering at them so that they take you seriously. Your lender will provide insights that you won't consider and are an excellent gauge of the deal as well. The lower the LTV they're wiling to lend to you at, the worse the deal is in their eyes.
Also pay close attention to management as a the quality of your manager can make or break your investment, unless you plan on self-managing. Always have a backup manager ready to go and don't assume that the current management is going to stay on. That said, figure out the managers motivations and see if it possible that they would want to stay on if you like the job they've done.
Obviously, don't forego your due diligence reports (pest, structural, nhd, plumbing, electrical, roof). Ask owner the condition and useful life of the big cap ex items and confirm with your own inspectors. You must factor those into your internal pro forma otherwise you run the risk of unforeseen expenses.
Hope that helps and good luck!