Does Respite foster care allow for deducting costs and do I include the state stipend on the 1099 as income although it is under $2000 for the year?

The stipends you received from the state for providing respite care are not taxable income.  So you would not report the stipends on your tax return.

 Per the IRS "You may be able to deduct as a charitable contribution some of the costs of being a foster parent"

 You can find this information by clicking here and look at page 5 of 22.  The information starts with "Foster Parents."

You can claim those amounts as a charitable contribution if the agency that placed the children with you can receive charitable contributions.  If they cannot, you will not be able to claim the amounts as a charitable contribution.