If the court is convinced that it would be in the best interest of the child to limit or restrict the father's interaction with the child, the court likely will order such restrictions. Whether or not there are restrictions, the father would still be ordered to pay child support.
As far as the child support issue, if you apply for public assistance, DSHS likely will start a proceeding to obtain a child support order.
If you do not apply for public assistance, DSHS will still help you get a child support order. The website for DSHS is http://www1.dshs.wa.gov/dcs/ .
In either case, DSHS will not get involved in the parenting issue (unless there are allegations that the child is being harmed).
WA LawHelp provides some self-help guides. Its website is http://www.washingtonlawhelp.org/WA/index.cfm . At the WA LawHelp site, you can search for low or no cost agencies that provide help.
Alternatively, you can retain a private attorney.