Do bed bug bites look like little bumps with pulse in it?

Staph versus bed bug. Bed bug bites are red bumps, but they don't usually have pus in them. They look more like a hive or mosquito bite than an infected boil. When pus is present in a skin lesion, i tend to think about staph infections. Those are common lately and seem to start anywhere they like. Fire ant bites can have pus in them, but they are sterile; it's due to the venom's damage. See your doctor for diagnosis.
Possibly. They are small but visible and if crushed after a blood meal will expell the blood just like a squashed mosquito.