Not sure where you live, But LSU Hospital in Shreveport, La is a very good Hospital that takes people with No Insurance...My parent's get Free care with Lsu, and it's been God sent for them...Hope you find a good place..
You ask an open-ended question. There is no such thing as "requesting an x-ray". How many views do you want and of what? And surprise - many facilities will tell you you don't own the x-rays and will refuse to them it to you. When this happened to me I just asked if "I could see them for a minute", put them under my arm and walked out of the building. The x-rays still have to be interpreted. An art rather than a science. As for blood tests, WHAT blood tests. There and dozens of them. As for an EKG, any paramedic ALS ambulance can perform a limited EKG, and all of the ones I know will do that for a good cheeseburger and coke for each of the crew members. In the United States there are thousands of clinics that will perform these services for minimal cost if you don't have means or insurance. One in my area charges a base fee of ten dollars per procedure.
If you are low income yeah the health dept can help you but their are some places that just do lab work and charge you 30-100 dlls for the tests, depends on what you want to check if your interested in a certain test like thyroid, or sugar, cholesterol then they can just test you for those particular things make some call and get estimates
Some tests are available through the Health Department. Call them to see what they provide free to start with.