I would budget separately. It makes it much easier to see how the business is actually performing and keeping track separately makes it easier to fill out your tax returns.
Not an answer to the exact question you asked, but along the same lines. If you have your personal and business funds in a shared account, I would encourage having a completely separate checking acct and debit and/or credit card for the rental/business.
It makes the accounting much easier, will make things simpler if the IRS audits you at some point, and for the deposit at least, may be required by law.
Requirements vary by state, but in general, you should not mingle the tenant's security deposit with your personal money. The deposit is the tenant's money (not yours) which you are holding in trust. You may be required to notify the tenant where the deposit is held, and you may be required to pay the tenant any interest earned in that account. It's worth looking into your state's regulations.