This one is a tough question because as the Westchestergov.com website puts it this is a county that "has had a shortage of fair and affordable housing." Having said that though, there are pockets of possibilities and it may take some research. It is worth noting that there are many communities/municipalities in Westchester that have adopted the Emergency Tenants Protection Act and so there are rules about a minimum number of rent control apartments. The communities and minimum no. of rent control units in each are as follows:
Croton (50)
Dobbs Ferry (6)
Eastchester (6)
Greenburgh (6)
Harrison (6)
Hastings (6)
Irvington (20)
Larchmont (6)
Mamaroneck Town (6)
Mount Kisco (16)
Mount Vernon (6)
New Rochelle (6)
Pleasantville (20)
Port Chester (12)
Sleepy Hollow (10)
Tarrytown (6)
White Plains (6)
Yonkers (6)
The state and county provide information about affordable housing:
http://www.nyhousingsearch.gov/http://homes.westchestergov.com/tenants/affordable-rental-aptsAs for your question about sliding scale, it looks like the county has community health centers in multiple locations that are similar to the Peekskill Center you have mentioned. According to
http://health.westchestergov.com/community-health-centers the following locations provide health care for those who do not have insurance:
Mount Vernon Neighborhood Health Center
Sites in Mount Vernon, Yonkers, Greenburgh
Hudson River HealthCare
Sites:
Peekskill
1037 Main Street, Peekskill, NY 10566
For an appointment call: (914) 734-8800
Yonkers - Park Care
2 Park Ave, Yonkers, NY 10703
For an appointment call: (914)964-7862
Yonkers - Valentine Lane
503 South Broadway, Yonkers, NY 10705
For an appointment call: (914)965-9771
Open Door Family Medical Centers
Mt. Kisco Open Door
30 West Main Street
Mt. Kisco, New York 10549
Phone: (914) 666-3272
Ossining Open Door
165 Main Street
Ossining, NY 10562
Phone: (914) 941-1263
Port Chester Open Door
5 Grace Church Street
Port Chester, NY 10573
Phone: (914) 937-8899
Sleepy Hollow Open Door
80 Beekman Avenue
Sleepy Hollow, NY 10591
Phone: (914) 631-4141