A small cut is not a hazmat incident. A bedroom after a hemorrhage is. Here is where the line sits.
For a small, fresh spill on a hard, non-porous surface (a cut finger's worth on tile or sealed wood), a careful adult can handle it: wear disposable gloves; wipe up visible blood with paper towels; disinfect the area with an EPA-registered disinfectant or fresh 1:10 household bleach solution and let it sit wet for several minutes; bag the towels and gloves, tie the bag, and put it in regular trash (small household quantities are generally acceptable there — sharps are not; they need a rigid container). Wash hands thoroughly afterward. If you have broken skin on your hands, let someone else do it.
Blood wicks. What looks like a stain on carpet is usually saturation through to the pad and often the subfloor — surfaces you cannot reach with a spray bottle. Professionals cut out and lawfully dispose of what cannot be disinfected, then verify the rest. If you are staring at a soaked mattress or carpet and wondering, the honest answer is: that is past the DIY line. See how to pick a company and what it should cost.
A fresh 1:10 bleach solution is effective on hard, non-porous, pre-cleaned surfaces with adequate wet contact time. It does not sterilize porous materials, and it damages many surfaces — it is a small-spill tool, not a remediation method.
Small household quantities (bandages, a few paper towels), generally yes. Saturated carpet, bedding, or large volumes are regulated medical waste in most states and need professional disposal.
Wash immediately with soap and water; flush eyes or mucous membranes with water. Then call your doctor or an urgent care promptly — post-exposure protocols for hepatitis B and HIV are time-sensitive.