If I am taking the standard deduction do I need to include the home I rent out (the mortgage interest as well as the rent income? I dont qualify to itemize.

Rental income is a form of business income, and has nothing what-so-ever to do with your standard deduction. All rental income and expenses are reported on SCH E as a part of your personal tax return, and deductible rental expenses have nothing to do with your standard deduction, or itemized deductions on the SCH A.

So if you have say $1000 in property taxes and renting our 35% of your residence, do you write off $350 on schedule e to offset rental income and also get to claim the standard deduction?

Your rental income, expenses and depreciation have nothing what-so-ever to do with your SCH A itemized deductions in any way, shape, form or fashion. All rental activity is reported on SCH E. Nothing concerning the rental activity goes anywhere else on your return.