Are dialysis patients with HIV allowed to be in the same dialysis room as other patients?

Yes. Hiv patients are not segregated from the other patients. All of the dialysis personnel use "universal precautions" and there is minimal risk of any blood contamination.
Absolutely. Hiv is spread via blood transfusion, sexual intercourse and intravenous drug use. It can not be transmitted during a dialysis treatment. There are many HIV positive patients who are on dialysis that sit next and touch non-hiv individuals who are also on dialysis and nothing happens.